Book of Abstracts - Dipartimento di Matematica

Workshop on
Non-Standard Numerical Methods for PDE’s
Pavia (Italy), June 29th - July 2nd, 2010
Book of Abstracts

Organizing Commetee
Daniele Boffi
Annalisa Buffa
Carlo Lovadina
Ilaria Perugia
Giancarlo Sangalli
Funding
Istituto di Alta Matematica “F. Severi”
cofounded by:
ERC Project GeoPDEs
Dipartimento di Matematica “F. Casorati”,
Università di Pavia

Program
Tuesday Wednesday Thursday Friday
9:00-9:50 D. N. Arnold P. Monk F. Brezzi E. Cohen
9:50-10:20 M. Costabel T. Betcke G. Manzini B. Jüttler
Break Break Break Break
10:50-11:20 S. Christiansen T. Luostari M. F. Wheeler A. Reali
11:20-11:50 J. Schoeberl G. Gabard I. Yotov J. A. Evans
11:50-12:20 R. Falk J. M. Melenk
Lunch
L. Beirão da
Veiga
J. Rivas
14:00-14:50 L. Demkowicz R. Hiptmair K. Lipnikov T. Dokken
14:50-15:20 F. Gardini A. Moiola P. Bochev C. Manni
15:20-15:50 R. Winther S. A. Sauter
15:20-15:40
D. Svyatskiy R. Duvigneau
15:40-16:00
A. Cangiani
Break Break Break
16:00-16:20
A. Quaglino
16:10-16:30 M. Dauge C. Hofreither F. Cirak
Break
16:30-16:50 H. Heumann J. Phillips P. N. Nielsen
16:50-17:10 A. Palha A. M. Marica A. Ratnani
17:10-17:30 J. Kreeft M. Ruess
16:40-20:00
Panel Session C. Heinrich
17:30-17:50 P. Gatto
20:30
Social dinner

Contents
Plenary lectures 1
Doug Arnold
Hilbert Complexes and the Finite Element Exterior Calculus . . . . . . . 3
Franco Brezzi
Scalar products and reconstructions in MFD . . . . . . . . . . . . . . . . 4
Elaine Cohen
Analysis Aware Modeling for Isogeometric Analysis . . . . . . . . . . . . . 5
L. Demkowicz, J. Bramwell and W. Qiu
Solution of dual-mixed elasticity equations using Arnold-Falk-Winther element
and discontinuous Petrov-Galerkin method, a comparison . . . . . 6
Tor Dokken
Locally Refined Splines . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
R. Hiptmair, C. Gittelson, A. Moiola, and I. Perugia
Plane Wave Discontinuous Galerkin Methods . . . . . . . . . . . . . . . . 9
K. Lipnikov
Mimetic finite difference method for solving PDEs on polygonal and polyhedral
meshes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Peter Monk
The solution of time harmonic wave equations using complete families of
elementary solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
Invited talks 13
L. Beirão da Veiga
Mimetic discretization of the diffusion problem: a higher order method
and a posteriori error estimates . . . . . . . . . . . . . . . . . . . . . . . . 15
Timo Betcke
Nonpolynomial FEM with MPSPACK . . . . . . . . . . . . . . . . . . . . 16
Pavel Bochev
Optimization-based computational models, or how to let someone else do
the hard work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
Snorre Christiansen
Mixed finite elements as inverse systems of differential forms . . . . . . . . 18
Martin Costabel
Regularized Poincaré operator and p version approximation of the Maxwell
eigenvalue problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

R. Duvigneau
Aerodynamic Analysis and Design using an Isogeometric Approach . . . . 20
Annalisa Buffa, John A. Evans, Thomas J.R. Hughes, Giancarlo Sangalli
Isogeometric Analysis in Fluid Dynamics: Divergence-Free Discretizations
for the Stokes and Navier-Stokes Equations . . . . . . . . . . . . . . . . . 21
R. Falk
Canonical Families of Finite Element Differential Forms and Their Properties
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
G. Gabard, R.J. Astley, P. Gamallo, G. Kennedy
A brief review of wave-based computational methods for aero-acoustics . . 23
Francesca Gardini
Mimetic finite differences for eigenvalue problems . . . . . . . . . . . . . 24
Bert Jüttler
Dimensions and Bases of Bivariate Hierarchical Tensor-product Splines . 26
Teemu Luostari, Tomi Huttunen and Peter Monk
A 3D plane wave basis for elastic wave problems . . . . . . . . . . . . . . 27
Carla Manni, Francesca Pelosi and Maria Lucia Sampoli
Beyond NURBS: Non-Standard CAGD Tools in Isogeometric Analysis . . 29
G. Manzini
High-order accurate nodal formulation of the Mimetic Finite Difference
Method for Elliptic Problems . . . . . . . . . . . . . . . . . . . . . . . . . 30
J.M. Melenk
Mapping properties of Helmholtz integral operators and their application
to the hp-BEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
Ralf Hiptmair, Andrea Moiolaand Ilaria Perugia
Approximation by plane waves . . . . . . . . . . . . . . . . . . . . . . . . 32
Alessandro Reali
Isogeometric Analysis in Pavia . . . . . . . . . . . . . . . . . . . . . . . . 34
L. Beirão da Veiga, A. Buffa, J. Rivasand G. Sangalli
Some estimates for hpk-refinement in Isogeometric Analysis . . . . . . . . 35
Stefan A. Sauter
A Posteriori Error Estimation for Highly Indefinite Helmholtz Problems . 36
Joachim Schöberl
Nitsche-type Mortaring for Maxwell’s Equations . . . . . . . . . . . . . . 37
Mary F. Wheeler
A Convergent MFMFE for Highly Distorted Hexahedra . . . . . . . . . . 38
R. Winther
Cochain projections in finite element exterior calculus . . . . . . . . . . . 39
Ivan Yotov
A Multiscale Mortar Multipoint Flux Mixed Finite Element Method . . . 40
Contributed talks 41

Andrea Cangiani
Mimetic Finite Difference methods for convection-diffusion problems . . . 43
Fehmi Cirakand Quan Long
A unified subdivision discretization approach for thick and thin shells . . 44
Monique Dauge
Skin effect in electromagnetism . . . . . . . . . . . . . . . . . . . . . . . . 45
Paolo Gatto, Leszek Demkowicz
Modeling of bone conduction of sound in the human head using hp-finite
elements: code design and verification. . . . . . . . . . . . . . . . . . . . . 47
Ch. Heinrich, A.-V. Vuong, B. Simeon
Finite Volume Methods on NURBS Geometries with Application to Fluid
Flow and Fluid-Structure Interaction . . . . . . . . . . . . . . . . . . . . . 48
Holger Heumann
High order Galerkin Methods for General Advection Problems . . . . . . 49
Clemens Hofreither
A non-standard finite element method based on boundary integral operators 51
J.J. Kreeft, A. Palha, M.I. Gerritsma
Higher-order discretization of the Laplace-Beltrami operator . . . . . . . . 52
Aurora-Mihaela Marica(joint work with Enrique Zuazua)
Localized solutions and filtering mechanisms for the discontinuous Galerkin
semi-discretizations of the 1 − d wave equation . . . . . . . . . . . . . . . 53
Peter Nørtoft Nielsen, Manh Dang Nguyen, Anton Evgrafov,
Allan Roulund Gersborg and Jens Gravesen
Isogeometric Shape Optimization . . . . . . . . . . . . . . . . . . . . . . . 55
A. Palha, J.J. Kreeft, M.I. Gerritsma
Higher order cochain interpolation . . . . . . . . . . . . . . . . . . . . . . 56
Joel Phillips
Quadrature for Finite Elements on Pyramids . . . . . . . . . . . . . . . . 57
Joachim Linn, Alessio Quaglino, Max Wardetzky and Clarisse Weischedel
Towards a Mimetic Nonlinear Shearable Shell . . . . . . . . . . . . . . . . 58
Ahmed Ratnani, Eric Sonnendrucker and Nicolas Crouseilles
High-Order Spline Finite Element Solver for the Time Domain Maxwell
equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
M. Ruessand E. Rank
A Fictitious Domain Method for Dynamic Analyses . . . . . . . . . . . . 62
D. Svyatskiy
Analysis of the Discrete Maximum Principle in the Mimetic Finite Difference
Method for Elliptic Problems . . . . . . . . . . . . . . . . . . . . . . 64
List of participants 65

Plenary lectures
1

Plenary lectures
Hilbert Complexes and the Finite Element Exterior Calculus
Doug Arnold
Institute for Mathematics and its Applications, University of Minnesota,
Minneapolis, MN 55455, USA
arnold@ima.umn.edu
The finite element exterior calculus (FEEC) provides a framework in which to understand
and develop stable finite element methods for a variety of problems. Hilbert
complexes provide a natural framework for FEEC. In this talk we will survey the basic
theory of Hilbert complexes and their discretization, discuss their applications to finite
element methods, and draw connections to results from geometry and topology.
3

Plenary lectures
Scalar products and reconstructions in MFD
Franco Brezzi
Istituto di Matematica Applicata e Tecnologie Iinformatiche, CNR,
via Ferrata 1, 27100 Pavia, Italia
brezzi@imati.cnr.it
It is well known that in many cases one has a fair amount of freedom in the choice of
scalar products for Mimetic Finite Differences. The best use of such freedom in many
circumstances is still to be assessed. Here in particular we will discuss the cases in
which the MFD scalar product could be interpreted as a traditional scalar product in
conforming or nonconforming finite element spaces. The usual consistency requirements
for the scalar products will be related to the classical “patch test” used by engineers in
order to assess the convergence of a given finite element discretization.
4

Analysis Aware Modeling for Isogeometric Analysis
Elaine Cohen
School of Computing, University of Utah,
50 S. Central Campus Drive, 3190 MEB, Salt Lake City, UT, 84112, USA
cohen@cs.utah.edu
Plenary lectures
Isogeometric Analysis (IA) has been proposed as a methodology for bridging the gap
between Computer Aided Design (CAD) and Finite Element Analysis (FEA). While
CAD typically focuses on a boundary representation, FEA has focused on volumetric
representations that are low order (linear) approximations to the geometry. Creating
high quality representations for just one of these goals can be challenging. However,
proposed representations for IA must create parameterizations and elements suitable
for supporting both good geometric computations and have good qualities for this new
mode of analysis. This presentations discusses some of the desirable attributes, challenges
in moving from current representational and modeling methodologies, and some
initial modeling/reconstruction methodologies towards creating models that satisfy both
domains.
5

Plenary lectures
Solution of dual-mixed elasticity equations using Arnold-
Falk-Winther element and discontinuous Petrov-Galerkin
method, a comparison
L. Demkowicz, J. Bramwell and W. Qiu
Institute for Computational Engineering and Sciences, The University of Texas at Austin,
Austin, TX 78712, USA
demlow@ms.uky.edu
Construction of stable Finite Element (FE) discretizations for the dual-mixed system
of linear elasticity has been an area of research for three decades. The recent element
of Arnold, Falk and Winther [1] builds upon reproducing on the discrete level an exact
sequence for the elasticity equations (with weakly imposed symmetry) that provides a
basis for stability of the continuous problem. The situation is more complicated than
for the classical grad-curl-div sequence where the discrete spaces are accompanied by
the operators from the continuous setting. Construction of the discrete elasticity complex
involves not only definition of appropriate discrete polynomial spaces but also a
necessary modification of the operators that involve besides the grad-curl-div operator
purely algebraic operations. In the end, the proof of discrete stability boils down to the
construction of appropriate interpolation operators that make simultaneously commute
three particular diagrams related to the elasticity complex.
The AFW element has been recently extended to elements of variable order [6,7,8].
Whereas the generalization of polynomial spaces is standard, the construction of the
interpolation operators is not and rather technical. 2D numerical experiments confirm
the stability not only for h-refined meshes of variable order, but for hp-refinements as
well. Showing the independence of stability constants upon order p, however, remains
open.
The main promise of the recently introduced Discontinuous Petrov Galerkin (DPG)
method with optimal test functions is that it guarantees the discrete stability, if the underlying
continuous problem is well-posed. The method delivers the best approximation
in a problem-dependent energy (residual) norm [2,3,4]. Moreover, with an appropriate
choice of norm for the test space, one can deliver the best approximation in a norm of
choice, starting with the L 2 -norm [5,9].
The presentation will be devoted to a numerical comparison and illustration of the two
methods using 2D examples. After a brief introduction of both methodologies, we shall
use an L-shape domain problem to study the stability of both methods by comparing
the actual approximation error with the best approximation error in context of uniform
as well as h- and hp-adaptive refinements.
6

References
Plenary lectures
[1] D. N. Arnold, R. S. Falk, and R. Winther. Mixed finite element methods for linear elasticity
with weakly imposed symmetry. Mathematics of Computation, 76:1699-1723, 2007.
[2] L. Demkowicz and J. Gopalakrishnan. A class of discontinuous Petrov-Galerkin methods.
Part I: The transport equation. Comput. Methods Appl. Mech. Engrg., 2009. accepted, see
also ICES Report 2009-12.
[3] L. Demkowicz and J. Gopalakrishnan. A class of discontinuous Petrov-Galerkin methods.
Part II: Optimal test functions. Technical Report 16, ICES, 2009. Numer. Meth. Part. D.
E., in review.
[4] L. Demkowicz, J. Gopalakrishnan, and A. Niemi. A class of discontinuous Petrov-Galerkin
methods. Part III: Adaptivity. Technical Report 1, ICES, 2010.
[5] A.H. Niemi, J.A. Bramwell, and L.F. Demkowicz. Discontinuous Petrov-Galerkin method
with optimal test functions for thin-body problems in solid mechanics. Technical Report 13,
ICES, 2010.
[6] W. Qiu and L. Demkowicz. Mixed hp-finite element method for linear elasticity with weakly
imposed symmetry. Comput. Methods Appl. Mech. Engrg., 198:3682-3701, 2009.
[7] W. Qiu and L. Demkowicz. Mixed hp-finite element method for linear elasticity with weakly
imposed symmetry II: Curvilinear elements in 2D. Technical Report 6, ICES, 2010.
[8] W. Qiu and L. Demkowicz. Mixed hp-finite element method for linear elasticity with weakly
imposed symmetry III: 3D elements. Technical report, ICES, 2010, in preparation.
[9] J. Zitelli, I. Muga, L. Demkowicz, J. Gopalakrishnan, D. Pardo, and V. Calo. A class of
discontinuous Petrov-Galerkin methods. Part IV: Wave propagation problems. Technical
Report 17, ICES, 2010.
7

Plenary lectures
Locally Refined Splines
Tor Dokken
Department of Applied Mathematics, SINTEF ICT,
Forskningsveien 1, Oslo
tor.dokken@sintef.no
The central idea of isogeometric analysis is to replace traditional Finite Elements
by NonUniform Rational B-splines (NURBS) to provide accurate shape description and
higher order elements in analysis. Since the introduction of the idea by T. Hughes in 2005
excellent results have been obtained documenting the potential of isogeometric analysis.
However, it has also been demonstrated that NURBS do not support the local refinement
needed in efficient isogeometric analysis. T-splines offer some local refinement but have
been shown to be linearly dependent in specific configurations, a situation not acceptable
in analysis. Rather than adapting technology such as T-splines developed for Computer
Aided Geometric Design (CAGD), Locally Refined Splines (LR-splines) are developed
from the dual viewpoints of CAGD and analysis. T-splines represent the refinement
in the grid of control vertices, a natural approach in CAGD. However, this hides the
dimensionality of the spline space, gives refinement a fairly large footprint, and can in
certain situations trigger an unwanted growth in the number of coefficients. LR-splines
code the refinement directly in the parameterization of the spline space by insertion of
knot line segments. This approach gives refinement a small footprint, provides control of
the dimensionality of the spline space, and supplies a basis composed of tensor product
B-spline basis functions with different levels of refinement. The LR-spline basis also
satisfies the important partition of unity property without resorting to rational scaling
as in T-splines. The talk will report on the current status of the development of LRsplines,
and show the first examples of the flexible refinement of LR-splines.
8

Plane Wave Discontinuous Galerkin Methods
R. Hiptmair † , C. Gittelson † , A. Moiola † , and I. Perugia ‡
† Seminar for Applied Mathematics, ETH,
Rämistrasse 101, 8092 Zürich, Switzerland
‡ Dipartimento di Matematica, Università di Pavia,
Via Ferrata 1, 27100 Pavia, Italia
Plenary lectures
{hiptmair,gittelson,moiola}@sam.math.ethz.ch, ilaria.perugia@unipv.it
A natural idea to improve the accuracy of Galerkin methods for wave propagation
problems in the frequency domain is to incorporate qualitative information about the
oscillatory behavior of solutions into the trial spaces. For the homogeneous Helmholtz
equation −∇u − ω 2 u = 0 this idea naturally resorts to plane wave functions x →
exp(iωd · x), |d| = 1, and it has given rise to various plane-wave based methods. Prominent
examples of such methods are the plane wave partition of unity finite element
method [2], the ultraweak Galerkin discretization (UWVF) [4, 5], the discontinuous enrichment
method [1, 13], and the VTCR (variational theory of complex rays) approach
[12]. All perform fairly well in computations, see [8-10] for computational results for the
ultra-weak approach. This presentation will focus on plane wave discontinous Galerkin
methods (PWDG), of which the UWVF is a special incarnation, see [3, 6]. In the framework
of Trefftz-DG methods the formulation is derived and, subsequently, the available
theoretical results on convergence of the h-version and the p-version will be surveyed [6,
7], resorting to refined plane wave approximation estimates from [11], cf. the talk by A.
Moiola. Numerical results on the performance of PWDG will be discussed along with
the extension of convergence theory to electromagnetic wave propagation. An outlook
will address the potential of adaptive plane wave methods and Trefftz-DG methods in
for other singularly perturbed boundary value problems.
References
[1] M. Amara, R. Djellouli, and C. Farhat, Convergence analysis of a discontin- uous Galerkin
method with plane waves and lagrange multipliers for the solution of Helmholtz problems,
SIAM J. Numer. Anal., 47 (2009), pp. 1038–1066.
[2] I. Babuˇska and J. Melenk, The partition of unity method, Int. J. Numer. Methods Eng., 40
(1997), pp. 727–758.
[3] A. Buffa and P. Monk, Error estimates for the ultra weak variational formulation of the
Helmholtz equation, Math. Mod. Numer. Anal., 42 (2008), pp. 925–940. Published online
August 12, 2008, DOI 10.1051/m2an:2008033.
9

Plenary lectures
[4] O. Cessenat and B. Després, Application of an ultra weak variational formulation of elliptic
PDEs to the two-dimensional Helmholtz equation, SIAM J. Numer. Anal., 35 (1998), pp.
255–299.
[5] — , Using plane waves as base functions for solving time harmonic equations with the ultra
weak variational formulation, J. Computational Acoustics, 11 (2003), pp. 227–238.
[6] C. Gittelson, R. Hiptmair, and I. Perugia, Plane wave discontinuous Galerkin methods:
Analysis of the h-version, Math. Model. Numer. Anal., 43 (2009), pp. 297– 331.
[7] R. Hiptmair, A. Moiola, and I. Perugia, Plane wave discontinuous Galerkin methods for
the 2D Helmholtz equation: Analysis of the p-version, Report 2009-20, SAM, ETH Zürich,
Zürich, Switzerland, 2009. Submitted to SINUM.
[8] T. Huttunen, M. Malinen, and P. Monk, Solving Maxwell’s equations using the ultra weak
variational formulation, J. Comp. Phys., 223 (2007), pp. 731–758.
[9] T. Huttunen and P. Monk, The use of plane waves to approximate wave propagation in
anisotropic media, J. Computational Mathematics, 25 (2007), pp. 350–367.
[10] T. Huttunen, P. Monk, and J. Kaipio, Computational aspects of the ultra-weak variational
formulation, J. Comp. Phys., 182 (2002), pp. 27–46.
[11] A. Moiola, R. Hiptmair, and I. Perugia, Approximation by plane waves, Report 2009-27,
SAM, ETH Zürich, Zürich, Switzerland, August 2009.
[12] P. Rouch and P. Ladeveze, The variational theory of complex rays: A predictive tool for
medium-frequency vibrations, Comp. Meth. Appl. Mech. Engr., 15 (2003), pp. 3301–3315.
[13] R. Tezaur and C. Farhat, Three-dimensional discontinuous Galerkin elements with plane
waves and lagrange multipliers for the solution of mid-frequency Helmholtz problems, Int.
J. Numer. Meth. Engr., 66 (2006), pp. 796–815.
10

Plenary lectures
Mimetic finite difference method for solving PDEs on polygonal
and polyhedral meshes
K. Lipnikov
Los Alamos National Laboratory, Theoretical Division, MS B284
Los Alamos, NM, USA, 87545
lipnikov@lanl.gov
A successful discretization method inherits or mimics fundamental properties of the
PDE such as conservation laws, symmetries, positivity structures and maximum principles.
Construction of such a method is made more difficult when the mesh is distorted
so that it can conform and adapt to the physical domain and problem solution. The
talk is about one such method - the mimetic finite difference (MFD) method. The MFD
method can be applied for solving problems with full tensor coefficients on unstructured
polygonal and polyhedral meshes. These meshes may include arbitrary elements:
tetrahedrons, pyramids, hexahedrons, degenerated and non-convex polyhedrons, and
generalized polyhedrons.
The classical MFD method has many similarities with other low-order discretization
methods, in particular, with the low-order mixed finite element method. Both methods
try to preserve fundamental properties of physical and mathematical models. The essential
difference is that the MFD method uses only the surface representation of discrete
unknowns to build elemental stiffness and mass matrices. Since no basis functions (no
extension inside the mesh element) is required, practical implementation of the MFD
method is simple for complex elements. Existence (but not uniqueness!) of real basis
functions corresponding to stiffness and mass matrices is used only in the convergence
analysis.
The talk introduces basic elements of the MFD method. Its various generalizations
developed over the last 5 years will be presented in other talks at this session. As an illustration
of the mimetic methodology, I present construction of the MFD method for diffusion
problem on generalized polyhedral meshes (joint work with F.Brezzi, M.Shashkov
and V.Simoncini). I describe families of discrete MFD schemes, summarize known theoretical
results and verify them with numerical experiments.
11

Plenary lectures
The solution of time harmonic wave equations using complete
families of elementary solutions
Peter Monk
Department of Mathematical Sciences, University of Delaware
Newark, DE 19716, USA
monk@math.udel.edu
This presentation is devoted to discussing plane wave methods for approximating the
time-harmonic wave equation paying particular attention to the Ultra Weak Variational
Formulation (UWVF). This method is essentially an upwind Discontinuous Galerkin
(DG) method in which the approximating basis functions are special traces of solutions
of the underlying wave equation. In the classical UWVF, due to Cessenat and Després,
sums of plane wave solutions are used element by element to approximate the global
solution. For these basis functions, convergence analysis and considerable computational
experience shows that, under mesh refinement, the method exhibits a high order of
convergence depending on the number of plane wave used on each element. Convergence
can also be achieved by increasing the number of basis functions on a fixed mesh (or
a combination of the two strategies). However ill-conditioning arising from the plane
wave basis can ultimately destroy convergence. This is particularly a problem near a
reentrant corner where we expect to need to refine the mesh.
The presentation will start with a summary of the UWVF and some typical analytical
and numerical results for the Hemholtz equation. It may be that different basis functions
need to be used in different parts of the domain. I shall present some numerical results
investigating convergence on an L-shaped domain using singular Bessel functions near
the corner. An alternative, that also extends to 3D, is to use polynomial basis functions
on small elements. Using mixed finite element methods, we can view the UWVF as a
hybridization strategy and I shall also present theoretical and numerical results for this
approach.
Although neither the Bessel function or the plane wave UWVF are free of dispersion
error (pollution error) they can provide a method that can use large elements and small
number of degrees of freedom per wavelength to approximate the solution. Extensions
to Maxwell’s equations and elasticity will be briefly discussed. Perhaps the main open
problems are how to improve on the bi-conjugate gradient method that is currently used
to solve the linear system, and how to adaptively refine the approximation scheme.
12

Invited talks
13

Invited talks
Mimetic discretization of the diffusion problem: a higher
order method and a posteriori error estimates
L. Beirão da Veiga
Department of Mathematics “F.Enriques”, University of Milan,
Via Saldini 50, 20133 Milano, Italy
lourenco.beirao@unimi.it
The main characteristic of the Mimetic Finite Difference (MFD) method, when compared
to a more standard finite element approach, is that the basis functions related to
the discrete degrees of freedom are not explicitly defined. As a consequence, the operators
and other quantities appearing in the problem must be approximated by discrete
counterparts that satisfy finite dimensional analogs of some fundamental property. This
approach allows for a greater flexibility of the mesh and the possibility to mimic intrinsic
properties of the differential problem under study. In particular, general polyhedral (or
polygonal in 2 dimensions) meshes, even with non convex and non matching elements,
can be adopted. This talk covers two developements on the mimetic approximation of
the diffusion problem in mixed form.
The first one is a higher order version of the original approximation proposed by Brezzi
et al. in 2005, which guarantees an O(h 2 ) approximation both for pressures and fluxes. A
post-processing procedure allows morevoer to build a piecewise quadratic pressure with
improved approximation properties. We note that a modified consistency condition must
be adopted in order to mantain the order of accuracy also when the diffusivity tensor is
variable across the domain.
In the second part of the talk we introduce a local a-posteriori error estimator for the
method, which applies both to the low and the high order method. The error estimator is
shown to be both reliable and efficient with respect to an energy type norm involving the
post-processed pressure. Finally, the error indicator is combined with a simple adaptive
process and a set of numerical tests for the low order case is presented.
15

Invited talks
Nonpolynomial FEM with MPSPACK
Timo Betcke
Department of Mathematics, University of Reading, Whiteknights,
PO Box 220, Berkshire, RG6 6AX, United Kingdom
The understanding of nonpolynomial finite element methods for Helmholtz problems
has seen tremendous progress over recent years. A common feature of most implementations
is that they are based on meshes with element sizes of a few wavelengths
in diameter, having local approximation spaces consisting of plane waves or sometimes
Fourier-Bessel functions. This follows the philosophy of standard finite element methods
of having many elements and few basis functions per element, leading to large and sparse
matrices with relatively small dense blocks.
In this talk we describe implementations of nonpolynomial finite elements based on
the philosophy of using as few elements as possible and to achieve convergence by pure
p-refinement on each of these global elements, where the types of basis functions can
differ from one element to another. The goal is to use on each element basis functions
that locally guarantee fast exponential convergence. This global element approach leads
to moderately sized, essentially dense matrices.
We present several examples based on the freely available Matlab Toolbox MPSPACK
developed by Barnett and Betcke for the solution of Helmholtz problems using global
elements and various types of nonpolynomial basis functions. In particular, we show
how to implement a simple and fast nonpolynomial finite element method for scattering
from convex and nonconvex polygons, which gives exponentially accurate results.
16

Invited talks
Optimization-based computational models, or how to let
someone else do the hard work
Pavel Bochev
Applied Mathematics and Applications, Sandia National Laboratories 1
MS 1320, Albuquerque, NM 87185, USA
pbboche@sandia.gov
Discretization is a model reduction process which converts infinite dimensional mathematical
models into finite dimensional algebraic equations that can be solved on a
computer. Consequently, discretization is accompanied by inevitable losses of information
which can adversely affect the predictiveness of the discrete models. Compatible
and regularized discretizations control these losses directly by choosing suitable field
representations and/or by modifications of the corresponding variational forms. Such
methods excel in control- ling “structural” information losses responsible for the stability
and well-posedness of the discrete equations. However, they encounter considerable
difficulties in at least two cases: multi-physics models which combine constituent components
with fundamentally different mathematical properties, and loss of “qualitative”
properties such as maximum principles, monotonicity and local bounds preservation.
In this talk we show optimization ideas can be used to control externally information
losses which are difficult (or impractical) to manage directly in the discretization
process. This allows us to improve predictiveness of computational models, increase
robustness and accuracy of solvers, and enable efficient reuse of code. Two examples
will be presented: an optimization-based framework for multi-physics coupling, and an
optimization-based algorithm for constrained interpolation (remap). In the first case,
our approach allows to synthesize a robust and efficient solver for a coupled multiphysics
problem from simpler solvers for its constituent components. To illustrate the
scope of the approach we derive a robust and efficient solver for nearly hyperbolic PDEs
from standard, off-the-shelf algebraic multigrid solvers for the Poisson equation, which
by themselves cannot solve the original equations. The second example demonstrates
how optimization ideas enable design of high- order conservative, monotone, bounds
preserving remap and transport schemes which are linearity preserving on arbitrary
unstructured grids, including grids with polyhedral and polygonal cells.
This is a joint work with D. Ridzal , G. Scovazzi (SNL) and M. Shashkov (LANL)
17

Invited talks
Mixed finite elements as inverse systems of differential forms
Snorre Christiansen
CMA c/o Department of Mathematics, University of Oslo,
P.O. Box 1053 Blindern, NO-0316 Oslo, Norway
snorrec@math.uio.no
Conforming mixed finite elements for the grad, curl and div operators can be thought
of in terms of differential forms. More precisely we argue that they are inverse systems
of finite dimensional complexes of differential forms, i.e. special functors from the set
of geometric objects in the mesh (ordered by inclusion) to the category of differential
complexes (equipped with pull-back). The Galerkin spaces are then recovered from the
local data as inverse limits in the sense of category theory. We show how the theory of
mixed finite elements can be developed in this setting, which naturally accommodates
polyhedral meshes as well as non-polynomial functions.
18

Invited talks
Regularized Poincaré operator and p version approximation
of the Maxwell eigenvalue problem
Martin Costabel
IRMAR, Université de Rennes 1, Campus de Beaulieu,
35042, Rennes Cedex, France
martin.costabel@univ-rennes1.fr
The regularized Poincaré integral operators are a new tool in the analysis of differential
forms that have a variety of different applications. In particular, they allow the
construction of vector and scalar potentials of optimal regularity in terms of fractional
order Sobolev norms. This can be used to complete a crucial step in the proof of the
discrete compactness of the p version edge elements, and therefore of their spectrally
correct approximation of the Maxwell eigenvalue problem.
In the talk, these integral operators and their companions, the Bogovskiĭ integral
operators, are presented with their main properties. Their role for the p version edge
element approximation of the Maxwell eigenproblem is explained.
Collaboration with:
Daniele Boffi (Pavia), Monique Dauge (Rennes), Leszek Demkowicz (Austin), Ralf Hitmair
(Zürich), Alan McIntosh (Canberra).
References
[1] M. Costabel, A. McIntosh: On Bogovskiĭ and regularized Poincaré integral operators
for de Rham complexes on Lipschitz domains. Math. Z. 265, No 2 (2010), 297–320.
[2] D. Boffi, M. Costabel, M. Dauge, L. Demkowicz, R. Hiptmair: Discrete compactness
for the p-version of discrete differential forms. IRMAR-Preprint 09-39, Universite
Rennes 1 2009. HAL : hal-00420150 ; arXiv : 0909.5079
19

Invited talks
Aerodynamic Analysis and Design using an Isogeometric
Approach
R. Duvigneau
INRIA Sophia-Antipolis Méditerranée,
BP 93, 2004 route des lucioles 06902 Sophia-Antipolis, France
Automated design optimization procedures are now commonly used in aerodynamics,
for drag minimization exercises or target pressure reconstruction problems. However,
these procedures are often cumbersome, since they are achieved by coupling existing sophisticated
numerical tools, originating from Computer Aided Design (CAD), optimization
(descent or evolutionary methods), simulation (finite-element, finite-volume), grid
generation (structured or unstructured), etc. Moreover, these tools are based on different
geometrical representations, ranging form NURBS (Non-Uniform Rational B-Splines)
used in CAD, grids used by physical solvers, to ad-hoc design parameterizations (angles,
lengths, volumes, etc) often used in optimization. Hence, several geometrical transformations
are required in practice to couple these tools, that could have severe consequences:
loss of accuracy, generation of spurious noise, local optima, non-differentiability, etc.
To definitely overcome these difficulties, T. Hughes proposed a few years ago to consider
a design process entirely based on NURBS, in which the simulation is achieved
using parameterized surfaces and volumes instead of grids.
In this study, the application of isogeometric analysis to aerodynamic problems will
be explored. Especially, a variational approach is carried out using a NURBS basis to
define the geometry and solution fields for compressible Euler equations, yielding highorder,
adaptive and hierarchical numerical schemes. The differences with respect to
the classical approach are underlined and some examples are demonstrated, concerning
aerodynamic analysis and design problems.
20

Invited talks
Isogeometric Analysis in Fluid Dynamics: Divergence-Free
Discretizations for the Stokes and Navier-Stokes Equations
Annalisa Buffa, John A. Evans, Thomas J.R. Hughes, Giancarlo Sangalli
Institute for Computational Engineering and Sciences, University of Texas at Austin,
3517 North Hills Drive, #T301, Austin, TX, USA 78731
evans@ices.utexas.edu
In this talk, I discuss recently developed isogeometric discretizations for the Stokes and
Navier-Stokes equations. These discretizations are a smooth generalization of classical
Raviart-Thomas elements in two dimensions and Raviart-Thomas-Nedelec elements in
three dimensions. When coupled with a weak enforcement of slip boundary conditions,
these discretizations lead to provably inf-sup stable and high-order convergent numerical
methods. As such, they are a natural candidate for use in the simulation of Stokes or
Navier-Stokes flow. In fact, when applied to incompressible flows, these discretizations
provide pointwise divergence-free velocity fields. This is an especially useful property
for coupled flow-transport problems where mass conservation is of great importance.
Furthermore, these discretizations admit a natural decomposition of the velocity basis
into divergence-free and complementary sub-bases, a useful property for fast and robust
solution of the resulting discrete system of equations. Numerical examples will be shown
illustrating optimal convergence rates for model problems and the effectiveness of this
technology for general incompressible flows.
21

Invited talks
Canonical Families of Finite Element Differential Forms and
Their Properties
R. Falk
Department of Mathematics, Rutgers University,
110 Frelinghuysen Road, Piscataway, NJ 08854
falk@math.rutgers.edu
We consider two families of finite element differential forms defined on simplicial
meshes in R n . The first family, which we denote by PrΛ k , is based on the space Pr
of polynomials of degree less than or equal to r and is a generalization of the BDM and
second kind Nédélec elements, while the second, which we denote by P − r Λ k , is based on
a polynomial space between Pr−1 and Pr and is a generalization of the Raviart-Thomas
and first kind Nédélec elements. We consider a number of important properties of these
spaces and especially their connections.
22

Invited talks
A brief review of wave-based computational methods for
aero-acoustics
G. Gabard, R.J. Astley, P. Gamallo, G. Kennedy
ISVR, University of Southampton,
University Road, Southampton, SO17 1BJ, UK
gabard@soton.ac.uk
Several wave-based methods developed for predicting sound propagation in non- uniform
flows are reviewed. The basic principle of wave-based methods is to incorporate
some known properties of the underlying physics into the numerical model. For instance,
instead of using standard polynomials or Chebyshev polynomials to approximate the solution,
wave-based methods generally use canonical solutions such as Green’s functions
or plane waves to construct a local description of the solution. The methods described
in this paper include the Green’s function discretisation, the partition of unity finite
element method and the wave-based discontinuous Galerkin method. The principles
of these methods are described and their performance and shortcomings are discussed.
While the methods are intended to be used with strongly inhomogeneous propagation
media, the canonical solutions used as interpolating functions are only valid for uniform
coefficients. Another issue that will be discussed is that of mesh generation.
23

Invited talks
Mimetic finite differences for eigenvalue problems
Francesca Gardini
Università degli Studi di Pavia,
via Ferrata 1, 27100 Pavia, Italy
francesca.gardini@unipv.it
In the last two decades, a variety of mimetic discretizations for the numerical approximation
of Partial Differential Equations has been developed. These numerical methods
are designed to mimic some fundamental properties of the continuous problems through
the definition of discrete operators that satisfy discrete analogs of the fundamental relations
of vector and tensor calculus, e.g., Gauss-Green’s identities.
In this talk, we extend the mimetic finite difference method to the solution of eigenvalue
problems. For more general results and exhaustive bibliography we refer to [3].
The convergence analysis is set in the framework of the classical spectral approximation
theory for compact operators; see [1]. The correct spectral approximation, i.e., a good
approximation of eigenmodes and the absence of spurious modes, is thus obtained by
proving the uniform convergence of the discrete solution operators to the continuous one.
The uniform convergence is a direct consequence of a new a priori error estimate for the
source problem, which improves the original a priori error analysis in [2] by relaxing the
H 1 -regularity assumption for the forcing term. Here, we obtain an improved error bound
in terms of the L 2 -norm of the right-hand side of the source problem by reformulating
the latter using an elemental lifting operator. Numerical results confirming the optimal
behavior of the method are presented.
The talk is based on a joint work with Andrea Cangiani (Università di Milano Bicocca)
and Gianmarco Manzini (Istituto di Matematica Applicata e Tecnologie Informatiche -
CNR).
References
[1] Babuˇska I., and Osborn J. 1991 Eigenvalue Problems. In Handbook of Numerical Analysis
(ed. P. Ciarlet and J. Lions), vol. 2, pp. 641–787. North Holland.
[2] Brezzi F., Lipnikov K., and Shashkov M. 2005 Convergence of the mimetic finite difference
method for diffusion problems on polyhedral meshes. SIAM J. Numer. Anal. 43, 1872–1896.
24

Invited talks
[3] Cangiani A., Gardini F., and Manzini G. 2009 Convergence of the mimetic finite difference
method for eigenvalue problems in mixed form. Technical Report 31PV09/24/0, IMATI-CNR
(submitted to Comput. Methods Appl. Mech. Engrg.).
25

Invited talks
Dimensions and Bases of Bivariate Hierarchical Tensorproduct
Splines
Bert Jüttler
Institute of Applied Geometry, Johannes Kepler University,
Altenberger Str. 69, 4040 Linz, Austria
bert.juettler@jku.at
Due to their potential applications for isogeometric analysis, tensor-product spline
spaces with local refinability have recently become an active topic of research. Most
of the existing and ongoing work focuses on the so-called T-splines of Sederberg et al.
These spline spaces, however, possess certain limitations, in particular with respect to
the locality of refinement operations. The talk will recall the classical construction of
hierarchical tensor-product splines. In addition, it will discuss dimensions and bases of
hierarchically refined tensor-product spline spaces and it will show how do construct
bases which are, non-negative, locally supported, and form a partition of unity.
26

A 3D plane wave basis for elastic wave problems
Teemu Luostari † , Tomi Huttunen † and Peter Monk ‡
† Department of Physics and Mathematics, University of Eastern Finland,
Kuopio Campus, P.O. Box 1627, FIN-70211, Finland.
‡ Department of Mathematical Sciences, University of Delaware,
Newark, DE 19716, USA.
teemu.luostari@uef.fi
Invited talks
Elastic wave problems and the Navier equation arise in many practical applications
in engineering. However, the modeling of elastic waves is more challenging and time
consuming than the modeling of the acoustics since the solution consists of different
wave modes (e.g. P-, SH- and SV-wave components). These modes can propagate with
different speeds in a medium. Therefore dense meshes are often needed in the standard
finite element method (FEM) to model the problem accurately. We shall consider a
competitive method to FEM, the ultra weak variational formulation (UWVF), which is
a special form of the discontinuos Galerkin method (DGM), see [2,3]. The UWVF was
first introduced and developed to the Helmholtz equation and Maxwell’s equation by
Cessenat and Després [1]. The feasibility of the UWVF for the elastic wave problems in
2D was studied in [4].
The UWVF is a non-polynomial volume based method that uses plane wave basis
functions, i.e. physical basis functions. In the elastic-UWVF the plane wave basis
consists of pressure and shear waves. Therefore in the UWVF the number of basis
functions is chosen separately for different wave modes. Using the plane wave basis the
computational burden is reduced because the integrals encountered in the method can
be computed efficiently in closed form. However, the plane wave basis may produce an
ill-conditioned matrix system which deteriorates the accuracy.
We shall consider the UWVF for the 3D elastic wave problem (Navier equation). The
model problem is simple elastic wave propagation in a cubic domain. We investigate
the performance of the UWVF with different wave numbers and different combinations
of the P-, SH- and SV-wave basis function ratios. We shall show the convergence with
different number of basis functions and discuss the conditioning of the method.
References
[1] Cessenat O, Després B. Application of an ultra weak variational formulation of elliptic
PDEs to the two-dimensional Helmholtz problem. SIAM Journal of Numerical Analysis
1998; 35(1):255–299.
27

Invited talks
[2] Gabard G. Discontinuous Galerkin methods with plane waves for time-harmonic problems.
Journal of Computational Physics, 225(2):1961–1984, 2007.
[3] Huttunen T, Malinen M, Monk P. Solving Maxwell’s equations using the ultra weak variational
formulation. Journal of Computational Physics 2007; 223(2):731–758.
[4] Huttunen T, Monk P, Collino F, Kaipio JP. The ultra weak variational formulation for
elastic wave problems. SIAM Journal on Scientific Computing, 25(5):1717–1742, 2004.
28

Invited talks
Beyond NURBS: Non-Standard CAGD Tools in Isogeometric
Analysis
Carla Manni † , Francesca Pelosi † and Maria Lucia Sampoli ‡
† Department of Mathematics, Università di Roma “Tor Vergata”,
Via della Ricerca Scientifica 1, 00133 Roma, Italy
‡ Department of Mathematics and Computer Sciences, Università di Siena,
Pian dei Mantellini 44, 53100 Siena, Italy
manni@mat.uniroma2.it
NURBS-based isogeometric analysis can be seen as a superset of FEM and offers a
powerful tool in the area of numerical methods for differential problems. The NURBSbased
approach leads to remarkable results, not only in the context of structural analysis,
but also in advection dominated flow phenomena, often characterized by sharp layers
involving very strong gradients.
Nevertheless the rational model (NURBS) presents several drawbacks both from the
geometrical and the analytical point of view. Therefore our attention has been focused on
overcoming the problems of NURBS standards, analyzing Isogeometric Analysis schemes
with different bases, but still equipped with refinement properties, geometric features
and classical algorithms as degree elevation, knot insertion, differentiation formulas,
etc. . . Such spaces admit a representation in term of functions which are a natural extension
of polynomial B-splines so that they are properly referred to as generalized
B-splines.
In this talk, after a general presentation, we focus on suitable spaces of generalized Bsplines
whose distinguishing property is the ability to efficiently describe sharp variations
without introducing extraneous oscillations. Thus, they seem particularly well suited to
face advection dominated flow phenomena.
29

Invited talks
High-order accurate nodal formulation of the Mimetic Finite
Difference Method for Elliptic Problems
G. Manzini
Istituto di Matematica Applicata e Tecnologie Informatiche, CNR,
via Ferrata 1, 27100 Pavia, Italy,
Centro per la Simulazione Numerica Avanzata CeSNA, IUSS,
27100 Pavia, Italy
marco.manzini@imati.cnr.it
The nodal formulation of the Mimetic Finite Difference Method for two-dimensional
elliptic problems is based on a consistency condition that reproduces exactly the integration
by parts formula for linear polynomials. Increasing the degrees of the polynomials
for which a consistency condition holds is a possible way to achieve higher order formulations,
but requires some care in the proper formulation of the method and in its
implementation. Basically, a new kind of degrees of freedom, moments of polynomials,
are introduced in the mimetic formulation to represent the elemental integral originated
by the integration by parts formula. Likewise, the edge integrals requires new additional
nodes and high order quadrature formulas of Gauss-Lobatto type. For example, when
quadratic polynomials are considered to implement a formally third-accurate method we
need one more degree of freedom per cell and one more degree of freedom per edge.
From a theoretical standpoint, optimal convergence rate is proved in a mesh-dependent
H 1 norm for polynomials of any order for meshes with very general shaped elements.
Experimental results confirm this behavior and show the optimal convergence rate in
this norm for polynomials of degree up to five when the method is applied to meshes of
quadrilateral and (possibly non-convex) polygonal cells. On such meshes, we have also
an experimental indication of optimal behavior when approximation errors are measured
using the L 2 norm.
This is a joint work with L. Beirao da Veiga and K. Lipnikov.
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Invited talks
Mapping properties of Helmholtz integral operators and
their application to the hp-BEM
J.M. Melenk
Institut für Analysis und Scientific Computing (E101), Technische Universität Wien
Wiedner Hauptstraße 8-10, A-1040 Wien, Austria
melenk@tuwien.ac.at
For the Helmholtz equation (with wavenumber k) on analytic curves and surfaces Γ
we analyze the mapping properties of the single layer, double layer as well a combined
potential boundary integral operator. A k-explicit regularity theory for the single layer
and double layer potentials is developed, in which these operators are decomposed into
three parts: the first part is the single or double layer potential for the Laplace equation,
the second part is an operator with finite shift properties, and the third part is an
operator that maps into a space of piecewise analytic functions. For all parts, the kdependence
is made explicit. We also develop a k-explicit regularity theory for the
inverse of the combined potential operator A = ±1/2 + K ′ − ikV and its adjoint, where
V and K are the single layer and double layer operators for the Helmholtz kernel. Under
the assumption that A −1 L 2 (Γ)←L 2 (Γ) grows at most polynomially in k, the inverse A −1
is decomposed into an operator A1 : L 2 (Γ) → L 2 (Γ) with bounds independent of k and
a smoothing operator A2 that maps into a space of analytic functions on Γ. The kdependence
of the mapping properties of A2 is made explicit. We show quasi-optimality
(in an L 2 (Γ)-setting) of the hp-version of the Galerkin BEM applied to A under the
assumption of scale resolution, i.e., the polynomial degree p is at least O(log k) and
kh/p is bounded by a number that is sufficiently small, but independent of k. Under this
assumption, the constant in the quasi-optimality estimate is independent of k. Numerical
examples in 2D illustrate the theoretical results.
This work is joint with M. Löhndorf (Vienna).
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Invited talks
Approximation by plane waves
Ralf Hiptmair † , Andrea Moiola † and Ilaria Perugia ‡
† Seminar for Applied Mathematics, ETH,
Rämistrasse 101, 8092 Zürich , Switzerland
‡ Dipartimento di Matematica, Università di Pavia,
Via Ferrata 1, 27100 Pavia - Italy
hiptmair@sam.math.ethz.ch, moiola@sam.math.ethz.ch,
ilaria.perugia@unipv.it
The Trefftz methods are special finite element methods where the trial and test functions
are solutions of the underlying PDE in the interior of each element. In the case
of the homogeneous Helmholtz equation −∆u − ω 2 u = 0, plane waves (x ↦→ e iωx·d ) or
circular/spherical waves are usually used as basis functions.
One of the main steps in the convergence analysis of any of these methods is the proof
of a best approximation estimate for the trial space. The only results of this kind for
plane wave spaces available in literature (see [?] and [?]) are limited to two dimensional
domains and are not completely satisfactory.
In order to improve and generalize these estimates we proceed in two steps. In the
first one, we show how u can be approximated by the so-called generalized harmonic
polynomials, i.e. the circular (in two dimensions) and the spherical (in three dimensions)
waves:
x = re iψ ↦→ e ±ilψ
x
Jl(ωr), x ↦→ Yl,m |x|
jl(ω|x|), 0 ≤ |m| ≤ l ∈ N.
The main tool used is the Vekua transform, a bijective integral operator that maps
Helmholtz solutions into harmonic functions on the same domain. This allows to reduce
the problem to the simpler case of the approximation of harmonic functions by harmonic
polynomials.
In the second step we approximate these functions with plane waves. The link between
plane waves and circular/spherical waves is provided by the Jacobi-Anger expansion. We
truncate the expansion and give a bound on the solution of the linear system obtained;
in three dimensions this requires a careful choice of the propagating directions of the
plane waves.
We show that, for any solution u ∈ H K+1 (D) of the homogeneous Helmholtz equation,
D ⊂ R N starshaped with diameter h, N = 2, 3, it is possible to approximate u with a
linear combination of p plane waves of given directions {dk}k=1,...,p with error
inf
α∈C p
u − p
iωx·dk
k=1αke ≤ C h j,ω,D K+1−j q −λ(K+1−j) uK+1,ω,D 32

Invited talks
where the Sobolev norms are weighted with ω, the constant C depends on ωh (explicitly),
j, k, {dk} and the shape of D. The number p of plane waves is linked to the parameter q
as p = 2q + 1 in two dimensions and as p = (q + 1) 2 in three dimensions. The parameter
λ depends only on the shape of D: in two dimensions it can be chosen arbitrarily close
to 1 while in three dimensions we can only prove its positivity.
The proof of analogous bounds for Maxwell equations (curl curl u − ω 2 u = 0) is more
difficult because the Vekua operators are not appropriate for these equations. We use
Herglotz functions, vector spherical harmonics and a special Jacobi-Anger formula to
separate the vectorial plane/spherical waves that are solution of Maxwell equations from
the ones that are not divergence free. With these tools we can prove a h-estimate for
spherical waves with the same order of convergence proved for the Helmholtz case but
using only a few more basis functions.
References
[1] O. Cessenat and B. Despres, Application of an ultra weak variational formulation of elliptic
PDEs to the two-dimensional Helmholtz equation, SIAM J. Numer. Anal. 35 1 (1998), 255–
299.
[2] R. Hiptmair, A. Moiola and I. Perugia, Approximation by plane waves, SAM Report 2009-
27, ETH Zürich.
[3] J.M. Melenk, On Generalized Finite Element Methods, Ph.D. thesis (1995), University of
Maryland, USA.
33

Invited talks
Isogeometric Analysis in Pavia
Alessandro Reali
Structural Mechanics Department, University of Pavia,
Via Ferrata 1, 27100 Pavia, Italy
alessandro.reali@unipv.it
Isogeometric analysis (IGA), introduced by the pioneering work of Hughes et al. in
2005, can now be considered more than just a promising technique in the framework of
numerical methods for PDEs, with many groups around the world contributing to its
development and diffusion. Some researchers from Pavia started working on IGA from
its infancy and now an entire research group is active on this topic, studying problems
ranging from mathematical analysis to code generation, from element technology to
engineering applications in solid mechanics, fluid dynamics, electromagnetics, etc. The
aim of this talk is to show some of the results obtained by our research group within
this context.
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Invited talks
Some estimates for hpk-refinement in Isogeometric Analysis
L. Beirão da Veiga ∗ , A. Buffa † , J. Rivas ‡ and G. Sangalli §
∗ Dipartimento di Matematica “F. Enriques”; Via Saldini, 50, 20133 Milano, Italy.
† IMATI, CNR; Via Ferrata 1, 27100 Pavia, Italy
‡ Departamento de Matemáticas, Universidad del País Vasco-Euskal Herriko Unibertsitatea,
Barrio Sarriena s/n, 48940 Leioa (Bizkaia), Spain
§ Dipartimento di Matematica, Università di Pavia; Via Ferrata 1, 27100 Pavia, Italy
lourenco.beirao@unimi.it, annalisa@imati.cnr.it, judith.rivas@ehu.es,
giancarlo.sangalli@unipv.it
New results on the approximation properties of two-dimensional Non-Uniform Rational
B-splines (NURBS) spaces will be presented. More precisely, we will define a new
projection operator into certain spaces of NURBS and give arror estimates in Sobolev
norms which are explicit in the three discretization parameters: mesh size, h, degree
p and regularity k. We firstly construct a one-dimensional projector onto the space of
splines of degree p with continuous derivatives up to the order k − 1, with the restriction
that 2k − 1 ≤ p, and obtain the corresponding error estimates. The extension to twodimensional
spline approximations is done through a tensor product construction and,
finally, the projector onto a NUBRS space is defined by multiplication with the weight
and composition with the mapping associated to the considered NURBS space. These
results are a first step in the analysis of the approximation properties of NURBS spaces
in terms, not only of the mesh size, but also of the other discretization parameters.
However, the restiction over the regularity, k, still leaves the most interesting cases of
higher regularity (up to k = p) open.
35

Invited talks
A Posteriori Error Estimation for Highly Indefinite
Helmholtz Problems
Stefan A. Sauter
Institut für Mathematik, Universität Zürich,
Winterthurerstrasse 190, 8057 Zürich
stas@math.uzh.ch
In this talk, we will consider Galerkin discretizations of highly indefinite Helmholtz
problems. Standard a priori and a posteriori error estimates, typically, suffer from the
ill-conditioned behaviour of the discrete Galerkin operator for large wave-numbers k.
Recently, it has been shown that - by choosing the polynomial order p in an hp-finite
element space according to p = O(log(k)) - the optimal a priori error estimates are
preserved. The result is based on a new regularity theory which employs a wave-number
dependent frequency splitting of the solution. In this talk, we generalize this regularity
theory for the posteriori error estimation of highly indefinite Helmholtz problems.
This talk comprises joint work with Willy Dörfler.
36

Nitsche-type Mortaring for Maxwell’s Equations
Joachim Schöberl
Institute for Analysis and Scientific Computing, Vienna University of Technology,
Wiedner Hauptstrasse 8-10, Wien, Austria
joachim.schoeberl@rwth-aachen.de
Invited talks
We propose a new method for treating transmission conditions on non-matching
meshes. The basic method is a hybrid version of Nitsche’s method. By introducing
a scalar potential on the interface we obtain a robust method for the low frequency
limit. In order to simplify numerical integration, we use smooth B-spline basis functions
of Nedelec-type on the interface. We present numerical results for scalar equations and
Maxwell’s equations in frequency domain.
37

Invited talks
A Convergent MFMFE for Highly Distorted Hexahedra
Mary F. Wheeler
Center for Subsurface Modeling, The University of Texas at Austin,
1 University Station C0200, Austin, Texas 78712, USA
mfw@ices.utexas.edu
In this presentation, we develop a new mixed finite element method for elliptic problems
on general quadrilateral and hexahedral grids that reduces to a cell-centered finite
difference scheme. A special non-symmetric quadrature rule is employed that yields a
positive definite cell-centered system for the pressure by eliminating local velocities. In
addition, this scheme is shown to be accurate on highly distorted quadrilateral and hexahedral
grids. Theoretical and numerical results indicate first-order convergence for the
pressure and face fluxes. Extensions to parabolic and two phase flow in porous media
are discussed.
This work has been done in collaboration with G. Xue and I. Yotov.
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Invited talks
Cochain projections in finite element exterior calculus
R. Winther
Centre of Mathematics for Applications (CMA), University of Oslo,
CMA, P.O. Box 1053, Blindern, 0316 Oslo, Norway
ragnar.winther@cma.uio.no
We will focus on the role of cochain projections, i.e. projections which commute with
the exterior derivative, in finite element exterior calculus. In particular, we will discuss
the relations between various uniform bounds on such projections and the convergence
properties of the associated finite element methods for boundary value problems and
eigenvalue problems. Furthermore, a new construction of local cochain projections of
Clément type will be presented.
39

Invited talks
A Multiscale Mortar Multipoint Flux Mixed Finite Element
Method
Ivan Yotov
Department of Mathematics, University of Pittsburgh,
301 Thackeray Hall, Pittsburgh, PA 15260, USA
yotov@math.pitt.edu
We present a multiscale mortar multipoint flux mixed finite element method for second
order elliptic problems. The equations in the coarse elements (or subdomains) are
discretized on a fine grid scale by a multipoint flux mixed finite element method that
reduces to cell-centered finite differences on irregular grids. The subdomain grids do
not have to match across the interfaces. Continuity of flux between coarse elements is
imposed via a mortar finite element space on a coarse grid scale. With an appropriate
choice of polynomial degree of the mortar space, we derive optimal order convergence
on the fine scale for both the multiscale pressure and velocity, as well as the coarse
scale mortar pressure. Some superconvergence results are also derived. The algebraic
system is reduced via a non-overlapping domain decomposition to a coarse scale mortar
interface problem that is solved using a multiscale flux basis. Numerical experiments
are presented to confirm the theory and illustrate the efficiency and flexibility of the
method.
This is joint work with Mary F. Wheeler and Guangri Xue, the University of Texas
at Austin.
40

Contributed talks
41

Contributed talks
Mimetic Finite Difference methods for convection-diffusion
problems
Andrea Cangiani
Dipartimento di Matematica e Applicazioni, Università di Milano Bicocca,
via Cozzi 53, 20125 Milano, Italy
andrea.cangiani@unimib.it
We present recent results on the extension of the Mimetic Finite Difference method
to elliptic equations comprising convection terms. We consider mimetic discretizations
based on both the nodal and mixed formulation of the problem. These are low order
methods that extend, respectively, the linear and Raviart-Thomas finite element to general
polygonal meshes. The analysis relies on the existence of appropriate elemental
lifting operators and, in the case of the mixed formulation, on the connection between
the mimetic formulation and the lowest order Raviart-Thomas mixed finite element
method. Stabilization techniques in the convection-dominated regimes will also be presented.
Numerical experiments on general polygonal meshes, also including non-convex
shaped elements, confirm the theoretical results and indicate a superconvergence effects
in some particular cases.
This a joint work with G. Manzini and A. Russo.
43

Contributed talks
A unified subdivision discretization approach for thick and
thin shells
Fehmi Cirak and Quan Long
Department of Engineering, University of Cambridge,
Trumpington Street, Cambridge CB2 1PZ , UK
http://www-g.eng.cam.ac.uk/csml/
fc286@eng.cam.ac.uk
We present a unified model and an associated discretization scheme for thin (shearrigid)
and thick (shear-flexible) shells. The equilibrium equations in weak form are
derived following a standard semi-inverse approach. Thereby, any configuration of the
shell is assumed to be defined as
ϕ = x + ζ(n + w) with − t t
2 ≤ ζ ≤ 2
where ϕ is the position vector of a material point within the shell volume, x is the
position vector of a material point on the mid-surface, n is the unit normal to the midsurface,
w is a vector for allowing out-of-plane shear deformations and t is the thickness
of the shell. In the assumed kinematics, x and w are the independent variables and n is
a function of x. Importantly, in the thin limit (t → 0) the Kirchhoff-Love model with
|w| ≡ 0 is recovered.
The conforming finite element discretization of the shell equations with the assumed
kinematics requires C 1 -continuous shape functions. In the present method, smooth
subdivision shape functions are used for interpolating the mid-surface position vector
x and the shear vector w. In the thin-shell limit, the resulting finite elements do not
exhibit shear locking simply because the concurrent interpolation of x = 0 and w ≡ 0
with the same shape functions does not lead to any compatibility problems.
44

Skin effect in electromagnetism
Monique Dauge
IRMAR, Université de Rennes 1, Campus de Beaulieu,
35042, Rennes Cedex, France
monique.dauge@univ-rennes1.fr
Contributed talks
We consider the Maxwell equations in a domain made of two parts, dielectric and
conductor. We are interested in the behavior of solutions when the conductivity in the
conductor part is large.
In a first stage, we assume smoothness for the dielectric–conductor interface and we
exhibit a multiscale asymptotic expansion with profile terms rapidly decaying inside
the conductor. We measure this skin effect by introducing a skin depth function that
turns out to depend on the mean curvature of the boundary of the conductor. We
then confirm these asymptotic results by numerical experiments in various axisymmetric
configurations.
In a second stage, motivated by numerical experiments in cylindrical configurations,
we also investigate the case of nonsmooth interfaces.
We bring extension or complements to previous works [10,6,7,5].
From collaboration with: Victor Péron (Bordeaux, and soon, Pau), Gabriel Caloz
(Rennes), Erwan Faou (Rennes), and Clair Poignard (Bordeaux), see [9,3,1,4].
Computations done with Finite Element Library Mélina [8].
References
[1] G. Caloz, M. Dauge, V. Péron. Uniform estimates for transmission problems
with high contrast in heat conduction and electromagnetism. To appear in Journal
of Mathematical Analysis and Applications (2010). http://hal.archives-ouvertes.fr/hal-
00422315/en/http://hal.archives-ouvertes.fr/hal-00422315/en/.
[2] G. Caloz, M. Dauge, E. Faou, V. Péron. On the influence of the geometry on skin
effect in electromagnetism. (In preparation), 2010.
[3] M. Dauge, E. Faou, V. Péron. Comportement asymptotique à haute conductivité de
l’épaisseur de peau en électromagnétisme. C. R. Acad. Sci. Paris Sér. I Math. (348) (2010)
385–390.
[4] M. Dauge, V. Péron, C. Poignard. Asymptotic expansion for the solution of a stiff
transmission problem in electromagnetism with a singular interface. (In preparation), 2010.
45

Contributed talks
[5] H. Haddar, P. Joly, H.-M. Nguyen. Generalized impedance boundary conditions for
scattering problems from strongly absorbing obstacles: the case of Maxwell’s equations.
Math. Models Methods Appl. Sci. 18(10) (2008) 1787–1827.
[6] R. C. MacCamy, E. Stephan. Solution procedures for three-dimensional eddy current
problems. J. Math. Anal. Appl. 101(2) (1984) 348–379.
[7] R. C. MacCamy, E. Stephan. A skin effect approximation for eddy current problems.
Arch. Rational Mech. Anal. 90(1) (1985) 87–98.
[8] D. Martin. Mélina, bibliothèque de calculs éléments finis. Source code. http://anummaths.univ-rennes1.fr/melinahttp://anum-maths.univ-rennes1.fr/melina,
1990-2010.
[9] V. Péron. Modélisation mathématique de phénomènes électromagnétiques dans des
matériaux à fort contraste. PhD thesis, Université Rennes 1 2009. http://tel.archivesouvertes.fr/tel-00421736/fr/http://tel.archives-ouvertes.fr/tel-00421736/fr/.
[10] E. Stephan. Solution procedures for interface problems in acoustics and electromagnetics.
In Theoretical acoustics and numerical techniques, volume 277 of CISM Courses and
Lectures, pages 291–348. Springer, Vienna 1983.
46

Contributed talks
Modeling of bone conduction of sound in the human head
using hp-finite elements: code design and verification.
Paolo Gatto, Leszek Demkowicz
Inst. for Comp. Engineering and Sciences, University of Texas at Austin,
Austin, TX 78712, USA
We focus on the development of a reliable numerical model for investigating the boneconducted
sound in the human head. The main difficulty of the problem arises from the
lack of fundamental knowledge regarding the transmission of acoustic energy through
non-airborne pathways to the cochlea, i.e. the most sensitive organ of the middle ear.
A fully coupled model based on the acoustic/elastic interaction problem with a detailed
resolution of the cochlea region and its interface with the skull and the air pathways,
should provide an insight into this fundamental, long standing research problem. To this
aim, we have developed a 3-D finite element code that supports elements of all shapes –
tetrahedra, prisms, and pyramids – in order to fully capture the complicated geometry
of the problem. We have tested our code employing the case of a multilayered sphere,
for which an analytical solution in terms of Hankel’s functions is known.
47

Contributed talks
Finite Volume Methods on NURBS Geometries with Application
to Fluid Flow and Fluid-Structure Interaction
Ch. Heinrich, A.-V. Vuong, B. Simeon
Centre for Mathematical Sciences, Technische Universität München,
Fakultät für Mathematik (M2), Lehrstuhl für Numerische Mathematik, Boltzmannstraße 3 D,
85748 Garching bei München, Germany
heinrich@ma.tum.de
Isogeometric Analysis extends isoparametric finite elements to more general basis functions
such as B-splines and NURBS. In this way, it is possible to fit exact geometries
at the coarsest level of discretization and eliminate geometry errors from the very beginning.
In CFD simulations, however, the Finite Volume Method (FVM) is still the
method-of-choice. This contribution discusses approaches to incorporate the idea of an
exact boundary representation and NURBS geometries in connection with the FVM. In
particular, we transform the problem formulation to the parametric domain and analyze
the discretization of the arising integral terms. Furthermore, the convergence behaviour
is studied for numerical examples in the field of flow simulations. Finally we present a
partitioned Fluid-Structure Interaction (FSI) coupling algorithm combining the solver
described above with a structural solver based on Isogeometric Analysis solving the
equations of linear elasticity. In this context, we also discuss an approach to incorporate
a matching interface.
48

Contributed talks
High order Galerkin Methods for General Advection Problems
Holger Heumann
Seminar for Applied Mathematics, ETH Zürich
Rämistrasse 101, 8092 Zürich, Switzerland
Holger.Heumann@sam.math.ethz.ch
Co-ordinate free differential forms offer considerable benefits for the construction of
compatible discretizations. We adopt here the perspective of differential forms to derive
Galerkin schemes for general advection problems:
∗ω + ∗Lβω = f in Ω ⊂ R n ,
ı ∗ ω = ı ∗ ω0 on inflow boundary,
iβω = iβω0 on inflow boundary.
(3.1)
These are boundary value problems for unknown l-forms ω, 0 ≤ l ≤ n, in the domain
Ω. The symbol ∗ stands for the Hodge operator mapping a l-form to a (n − l)-form.
Boundary conditions on the inflow boundary are imposed via the trace ı ∗ ω and the
contraction iβω of ω. By Lβ we denote the Lie derivative for a given velocity field β.
The Lie derivative generalizes the concept of directional derivatives of scalar functions to
differential forms, hence it models advection. For n = 3 the vector analytic incarnations
of the Lie derivative for unknown fields u or functions u are:
• l=0: β · grad u;
• l=1: curl u × β + grad(u · β);
• l=2: βdivu + curl(β × u);
• l=3: div(βu).
While the cases l = 0 and l = 3 have been object of intense research, the cases l = 1 and
l = 2 have received scant attention, though they are relevant for numerical modeling in
e.g. symmetrized MHD [2] or magnetic diffusion convection problems [4].
By means of differential forms calculus we derive a consistent and stable Galerkin
discretization of (1) for general, not necessarily conforming, approximation spaces. For
discontinuous approximation spaces we even obtain stability in some mesh dependent
norm, which is stronger than the L 2 -norm. In the cases l = 0 and l = n these schemes are
49

Contributed talks
the usual Galerkin schemes for scalar advection and globally continuous or discontinuous
approximation spaces.
1
k+ We then prove for the case l = 1 and n = 3 optimal convergence of order O(h 2 )
for either fully discontinuous, normally continuous or tangentially continuous piecewise
polynomial approximation spaces. The proof is very much inspired from the corresponding
proof for l = 2, n = 2 [3] and the principle ideas can be extended to the curl- and
div-conforming approximation spaces due to the build-in finite element de Rham complex
[1].
References
[1] D.N. Arnold, R.S. Falk and R. Winther: Finite element exterior calculus, homological techniques,
and applications, Acta Numerica, vol. 15 (2006), 1-155.
[2] T. Barth: On the role of involutions in the discontinous Galerkin discretization of Maxwell
and magnetohydrodynamic systems, Compatible spatial discretizations, IMA Vol. Math.
Appl.,vol 142 (2006), 69-88
[3] F. Brezzi, L.D.Marini, E. Süli: Discontinuous Galerkin methods for first-order hyperbolic
problems, Math. Models Methods Appl. Sci., vol 14 (2004), 1893–1903
[4] Holger Heumann, Ralf Hiptmair and Jinchao Xu: A semi–Lagrangian method for convection
of differential forms, SAM-Report 2009-9, 2009.
50

Contributed talks
A non-standard finite element method based on boundary
integral operators
Clemens Hofreither
Institute of Computational Mathematics,
Altenberger Strasse 69, A-4040 Linz, Austria/Europe
clemens.hofreither@dk-compmath.jku.at
Joint work Ulrich Langer and Clemens Pechstein.
We present a non-standard finite element method based on the use of boundary integral
operators that permits polyhedral element shapes as well as meshes with hanging
nodes. The method employs elementwise PDE-harmonic trial functions and can thus be
interpreted as a local Trefftz method. The construction principle requires the explicit
knowledge of the fundamental solution of the partial differential operator, but only locally,
i.e. in every polyhedral element. This allows us to solve PDEs with elementwise
constant coefficients. In this talk we consider the diffusion equation as a model problem,
but the method can be generalized to convection-diffusion-reaction problems and to
systems of PDEs like the linear elasticity system with elementwise constant coefficients.
We provide a rigorous error analysis of the method for the three-dimensional case under
quite general assumptions on the geometric properties of the elements. Some numerical
tests confirm the theoretical results.
51

Contributed talks
Higher-order discretization of the Laplace-Beltrami operator
J.J. Kreeft, A. Palha, M.I. Gerritsma
Delft University of Technology
Kluyverweg 1, 2629 HS Delft, The Netherlands
J.J.Kreeft@tudelft.nl
The Laplace-Beltrami operator acting on differential forms is given by ∇ := δd + dδ :
Λ k ↦→ Λ k . This operator resembles among others the Laplace of scalars as well as the
Laplace acting on vectors. An example of the former is in the Poisson equation and the
later can be found e.g. in fluid dynamics and electro-magnetism [1].
In this talk we show how this operator fits in the double DeRham-complex [2] and the
two different ways to discretize/approximate the Laplace-Beltrami operator. The first is
in approximating the Hodge-⋆ operator, resulting in a staggered-grid finite-volume like
method, and the second is in approximating the dual DeRham-complex, resulting in a
(mixed) finite-element like formulation [3].
The presented formulations are combined with the mimetic spectral element method,
a higher-order interpolation method for cochains, the discrete counterpart of differential
forms. The robustness of the proposed method is demonstrated using highly curved
grids while still maintaining higher-order convergence.
References
[1] F. Brezzi, A. Buffa, Innovative mimetic discretizations for electromagnetic problems, Journal
of Computational and Applied Mathematics, 234, 1980-1987, (2010).
[2] M. Desbrun, E. Kanso, Y. Tongy, Discrete differential forms for computational modeling,
in proceedings of International conference on computer graphics and interactive techniques,
(2005).
[3] D.N. Arnold, R.S. Falk, R. Winther, Finite element exterior calculus: from Hodge theory
to numerical stability, Bulletin of the American mathematical society, 47, 281-354, (2010).
52

Contributed talks
Localized solutions and filtering mechanisms for the discontinuous
Galerkin semi-discretizations of the 1−d wave equation
Aurora-Mihaela Marica (joint work with Enrique Zuazua)
BCAM - Basque Center for Applied Mathematics,
Bizkaia Technology Park, 500, E-48160, Derio, Basque Country, Spain
marica@bcamath.org
We perform a complete Fourier analysis of the semi-discrete 1 − d wave equation
obtained through P1 discontinuous Galerkin (DG) approximations of the continuous
wave equation on an uniform grid. The resulting system exhibits the interaction of
two types of components: a physical one and a spurious one, related to the possible
discontinuities that the numerical solution allows. Each dispersion relation contains
critical points where the corresponding group velocity vanishes. Following previous
constructions in [5], we rigorously build wave packets with arbitrarily small velocity of
propagation concentrated either on the physical or on the spurious component. We also
develop filtering mechanisms aimed at recovering the uniform velocity of propagation of
the continuous solutions. Finally, some applications to numerical approximation issues of
control problems and connections to the discrete estimates for the Schrödinger equation
are presented.
Key words: wave equation, discontinuous Galerkin methods, Fourier analysis, dispersion
relation, group velocity, filtering, bigrid algorithm.
References
[1] D.N.Arnold, F.Brezzi, B.Cockburn, L.D.Marini, Unified analysis of Discontinuous Galerkin
Methods for Elliptic Problems, SIAM J. Numer. Anal., 39 (2002), 1749-1779.
[2] R.Glowinski, J.-L.Lions, J.He, Exact and approximate controllability for distributed parameter
systems: a numerical approach, Encyclopedia of Mathematics and its Applications,
Cambridge University Press, 117 (2008).
[3] L.Ignat, E.Zuazua, Convergence of a multi-grid method for the control of waves, J. Eur.
Math. Soc., 11 (2009), 351391.
[4] L.Ignat, E.Zuazua, Numerical dispersive schemes for the nonlinear Schrödinger equation,
SIAM. J. Numer. Anal., 47(2) (2009), 1366-1390.
[5] A.Marica, E.Zuazua, Localized solutions for the finite difference semi-discretization of the
wave equation, C.R. Acad. Sci. Paris, to appear.
53

Contributed talks
[6] A.Marica, E.Zuazua, Localized solutions and filtering mechanisms for the discontinuous
Galerkin semi-discretizations of the 1 − d wave equation, C.R. Acad. Sci. Paris, submitted.
[7] E.Zuazua, Exponential decay for the semilinear wave equation with localized damping in
unbounded domains, J. Math. Pures Appl., 70 (1991), 513-529.
[8] E.Zuazua, Propagation, Observation, Control and Numerical Approximations of Waves,
SIAM Review, 47(2)(2005), 197-243.
54

Isogeometric Shape Optimization
Peter Nørtoft Nielsen ∗† , Manh Dang Nguyen ∗ , Anton Evgrafov ∗ ,
Allan Roulund Gersborg † and Jens Gravesen ∗
∗ DTU Mathematics, Technical University of Denmark,
Bygning 303S, DK-2800 Kgs. Lyngby, Denmark
† DTU Mechanical Engineering, Technical University of Denmark
Nils Koppels Allé, Building 404, DK-2800 Kgs. Lyngby, Denmark
p.n.nielsen@mat.dtu.dk
Contributed talks
Isogeometric analysis unites the power to analyse complex engineering problems from
finite element analysis (FEA) with the ability to represent complicated shapes from
computer aided design (CAD)[1, 2]. Being a mixture of CAD and FEA, isogeometric
analysis serves as an ideal basis for shape optimization [1, 3, 4].
In the first part of the talk we apply the method of isogeometric shape optimization
to 2-dimensional, steady-state Stokes flow problems with Dirichlet boundary conditions.
We present implementation details and results for a simple optimization problem in
which the objective is to find the most energy efficient shape of a 2-dimensional pipe
bend.
In the second part, we present isogeometric shape optimization towards the problem
of designing vibrating membranes where their first few eigenfrequencies are prescribed.
Some nice shapes have been obtained by different methods. They thus show the robustness
of our approach and support the integration of isogeometric analysis into shape
optimization.
References
[1] T.J.R. Hughes, J.A. Cottrell, Y. Bazilevs, Isogeometric analysis: CAD, finite elements,
NURBS, exact geometry and mesh refinement, Comput. Methods Appl. Mech. Engrg., 194,
4135–4195, 2005.
[2] J.A. Cottrell, T.J.R. Hughes, Y. Bazilevs, Isogeometric Analysis. Toward Integration of
CAD and FEA, John Wiley and Sons, 2009.
[3] W.A. Wall, M.A. Frenzel, C. Cyron, Isogeometric structural shape optimization, Comput.
Methods Appl. Mech. Engrg., 197, 2976–2988, 2008.
[4] S. Cho, S.-H. Ha, Isogeometric shape design optimization: exact ge- ometry and enhanced
sensitivity, Struct. Multidisc. Optim, 38, 53–70, 2009.
55

Contributed talks
Higher order cochain interpolation
A. Palha ∗ , J.J. Kreeft † , M.I. Gerritsma †
∗ Delft University of Technology
Kluyverweg 1, 2629 HS Delft, The Netherlands
{a.palha,j.j.kreeft,m.i.gerritsma}@tudelft.nl
Algebraic topology can be considered as the natural discrete analogue of differential
geometry. This analogy is used in so-called mimetic discretization schemes. The kforms
from differential geometry have a natural discrete counterpart in the k-cochains
in algebraic toplogy. The exterior derivative, d, from differential geometry with all its
properties can be represented discretely by the coboundary operator, δ, with similar
properties. The DeRham map, R, maps k-forms onto k-cochains, whereas the Whitney
map, I, maps k-cochains onto k-forms. The DeRham map consists of integration of kforms
over oriented k-dimensional geometric objects, whereas the Whitney map consists
of interpolation, [1]. The maps should be constructed such that
R · I ≡ I and I· = O(hp), (3.2)
where I is the identity map, h is the characteristic mesh size and p the order of the
Whitney map. Furthermore, we require that
R · d = δ · R and d · I = I · δ. (3.3)
Given a cell complex, K, the DeRham map is trivial. This talk will present a Whitney
map of arbitrary order satisfying the above requirements. In addition, it will be shown
that if these interpolation functions are used as basis functions for finite element formulations
for conservation laws, well-known finite volume discretizations are obtained.
Since all operators are purely topological, all these relations remain valid in severely
distorted domains, [2].
References
[1] P.B. Bochev and J.M. Hyman, Principles of mimetic discretizations of differential equations,
IMA, 142, Eds. D. Arnold, P. Bochev, R. Nicolaides and M. Shashkov, (2006).
[2] M. Gerritsma, Edge functions for spectral element methods, in proceedings of ICOSAHOM
2009, (2010).
56

Quadrature for Finite Elements on Pyramids
Joel Phillips
Mathematics, Reading University,
Whiteknights, PO Box 220, Reading RG6 6AX, UK
joel.phillips@reading.ac.uk
Contributed talks
High order finite elements on pyramidal domains that conform to and commute with
the spaces of the de Rham sequence have recently been constructed. A practical implementation
of a finite element method requires a quadrature rule, of which the basic
requirement is that it not decrease the order of convergence of the method.
The analysis of the effect of quadrature on polynomial elements is classical. One
rule of thumb is that for the approximation of an elliptic problem using finite elements
consisting of polynomials of degree k, it is sufficient to use a quadrature rule that exactly
integrates all polynomials of degree 2k − 2. However, useful conforming pyramidal finite
elements cannot be constructed with purely polynomial approximation spaces - it is
essential to include certain rational functions.
Quadrature rules for pyramids based on conical product formulae that exactly integrate
polynomials of a given degree have been known for over half a century. Promisingly,
these rules also exactly integrate the rational functions included in each of the pyramidal
finite element families. However, the classical theory requires that the approximation
space is a subspace of H k (K) for each element K. This condition is not satisfied by the
rational functions present in the pyramidal elements and so the classical rule of thumb
does not immediately generalise.
In this talk, we shall show how to overcome this problem via a refinement of the
Bramble-Hilbert lemma.
57

Contributed talks
Towards a Mimetic Nonlinear Shearable Shell
Joachim Linn ∗ , Alessio Quaglino † , Max Wardetzky † and Clarisse Weischedel †
∗ Fraunhofer Institute for Industrial Mathematics, Kaiserslautern, Germany,
† Department of Numerical and Applied Mathematics, University of Göttingen, Germany
quaglino@math.uni-goettingen.de
Linear shearable plates have received wide attention due to the effort of deriving compatible
finite element discretizations for the Reissner-Mindlin model [1], a topic which
by now enjoys a mature literature. The inherent kinematic assumptions, allowing for
transverse shear strain, have been successfully employed in the geometrically nonlinear
setting, leading to the well-known Naghdi shell model.
The fact that the FE view ofttimes occludes the geometry behind a given physical
model has stirred recent interest in discrete geometric models, e.g., grounded in exterior
calculus [2]. By incorporating a geometric viewpoint for the case of shells, we propose a
mimetic FD discretization, with the aim to provide a unified language for geometrically
linear and nonlinear deformations of shells and plates. We argue that this goal can be
achieved by discretizing the geometric quantities underlying the smooth theory in an
invariant, parameterization-free manner.
We present in detail one particular triangular element that is derived from these
considerations. We show that this discretization offers an independent derivation of a
shell element proposed by Flores et al. [3] and generalizes a previously considered discrete
thin-shell model by Grinspun et al. [4]. We further argue that any low-order finite
element for the Reissner-Mindlin model can be translated into our geometric language.
Additionally, we provide a geometric intuition for shear locking.
Overview
In contrast to the the Naghdi description of shells, the geometric view describes curvatures
by the change of normals instead of derivatives of positions. Within this framework,
moderately thick shells may be treated by the Cosserat model [5] that allows for midsurface
“normals”—referred to as directors—to undergo transverse shear, i.e., to deviate
from true midsurface normals. By attaching directors to surfaces, the Cosserat model
immediately lends itself to a differential-geometric treatment, based on the theory of
Cartan’s moving frames. If the directors coincide with surface normals, it recovers the
Koiter model of thin shells.
58

Contributed talks
We represent the first and second fundamental forms of the deformed surface, encoding
intrinsic metric and extrinsic curvature properties, respectively, by
I = dφ T dφ, II = 1
dd
2
T dφ + dφ T
dd , (3.4)
where φ : ¯ S → R3 denotes the (orientation preserving) deformation of the midsurface,
d : ¯ S → R3 denotes the director field, and d denotes the (metric-free) Cartan outer
derivative. To get a low-order model, we choose φ to be a P1 -mapping between the
undeformed and deformed configuration, so that dφ (and therefore the first fundamental
form) is readily obtained in the discrete case. A Cosserat shell’s elastic potential energy
is the sum over membrane, bending, and shearing contributions:
W = t
2
¯S
I − Ī2 Km + t2 II − ĪI2 Kb + ksd tan 2 . (3.5)
Here Ī, ĪI denote the fundamental forms of the undeformed surface, respectively, and
dtan is the director’s tangential component with respect to the deformed surface. Further,
· K denotes a norm encoding material stiffnesses, respectively for membrane and
bending, and ks represents the shearing stiffness, which can be treated as a scalar in the
isotropic case. While our invariant energy formulation is equivalent to a parameterizationdependent
description considered by, e.g., Simo and coworkers [6], we are not aware of a
similarly compact representation as ours—a formulation that almost immediately carries
over to the discrete setting.
For a discrete treatment of shear, we assign surface normals and directors to edge
degrees of freedom of a triangulated surface mesh. While it might appear that discrete
surface normals are naturally defined as angle-bisecting ones, we present numerical evidence
that this choice leads to shear locking when the discrete mesh interpolates a given
smooth surface. Instead, we allow the unit midedge normal to adjusts itself according
to minimizing the elastic energy, and show that this choice is equivalent to reduced
integration in the FE language.
For obtaining a discrete second fundamental form II, we observe that dd is a (multivalued)
1-form in the language of exterior calculus. In the discrete view, dd is an
integrated quantity, which can be applied to 1-dimensional cells. We choose midedge
connecting lines as such requisite 1-cells, giving three directions per triangle for which II
can be evaluated. Fortunately, in dimension two, any quadratic form (and hence II) is
uniquely determined by its action on three different vectors, yielding a constant bending
strain per triangle, which can be expressed in a similarly compact and invariant manner
as the strain of a constant strain triangle.
References
[1] F. Brezzi, M. Fortin, Mixed and Hybrid Finite Element Methods, Springer-Verlag, 1991.
[2] D. Arnold, R. Falk, R. Winther, Finite Elements Exterior Calculus, Acta Numerica, 2006.
[3] F. G. Flores, E. Oñate, F. Larate, New assumed strain triangles for non linear shell analysis,
J. Comp. Mech., pp. 107-114, Vol. 17, 1995.
59

Contributed talks
[4] E. Grinspun, Y. Gingold, J. Reisman, D. Zorin, Computing discrete shape operators on
general meshes, Eurographics (Com- puter Graphics Forum), Vol. 25, 2006.
[5] M. Bischoff, W. A. Wall, K.-U. Bletzinger, E. Ramm, Models and Finite Elements for Thinwalled
Structures, Encyclopedia of Computational Mechanics Vol. 2, Wiley, 2004.
[6] J. C. Simo, D. D. Fox, On Stress Resultant Geometrically Exact Shell Model. Part I: Formulation
and Optimal Parametrization, J. Comp. Meth. Appl. Mech. Eng., pp. 267-304, Vol.
72, 1989.
60

Contributed talks
High-Order Spline Finite Element Solver for the Time Domain
Maxwell equations
Ahmed Ratnani ∗ , Eric Sonnendrucker † and Nicolas Crouseilles ∗
∗ INRIA and University of Strasbourg, France,
† University of Strasbourg, France
ahmed.ratnani@math.unistra.fr
Isogeometric analysis has been developed recently in order to be able to use the CAD
description of the computational domain to define Finite Element basis functions. In the
same spirit, we developed an exact sequence of Finite Element spaces based on splines
having the same properties as the Whitney Finite Element spaces traditionnally used
for the Finite Element solution of Maxwell’s equations. As with the Whitney elements,
one of Ampere’s or Faraday’s law can be discretized with a relation between the spline
coefficients of the eletric and magnetic fields independent of the topology of the mesh.
The metric comes in through a discrete Finite Element Hodge operator which appears as
the mass matrix involved in the other equation. All domains represented by NURBS can
be represented exactly and arbitrary high degree splines can be used. We shall describe
a Time Domain Maxwell solver based on these Finite Elements and a few validation test
cases.
References
[1] Ratnani, Sonnendrucker, Crouseilles. Arbitrary High-Order Spline Finite Element Solver for
the Time Domain Maxwell equations (In preparation)
61

Contributed talks
A Fictitious Domain Method for Dynamic Analyses
M. Ruess and E. Rank
Center for Simulation Technology in Engineering - CeSIM, Technische Universität München,
Arcisstrasse 21, 80333 München, Germany
ruess@tum.de
The analysis of the dynamic response of complex structures from solid mechanics often
requires an essential abstraction and simplification of the true structural properties due
to a tremendous discretization effort and often is limited with traditional methods.
With the Finite Cell Method (FCM) [1, 2], a novel high-order fictitious domain method
was proposed that fully embeds the structure in an extended, coarse orthogonal cell
structure and identifies its true geometry on the numerical integration level. The method
has been intensively studied for linear analyses and recently showed very promising
results also for nonlinear analyses [3]. Dynamic analyses, however, haven’t yet gained
much attention and will be thoroughly examined in this contribution.
The Finite Cell Method dramatically simplifies the modelling effort for complex geometries
and/or multi-material interfaces and essentially keeps the advantageous properties
of the p-version of the Finite Element Method, thus combining high accuracy, exponential
convergence rates and predictable error properties. Even more, the method turns out
to be highly suited for non-homogeneous material models where the constitutive properties
change continuously within the computation domain. The quality of the solution
for the proposed method depends on several factors, such as the numerical integration
scheme and a balanced h/p refinement strategy that will be discussed and analysed with
regard to dynamic structure response.
The numerical integration scheme requires a dense distribution of integration points
to accurately capture the structural inhomogeneity and to confine the integration error.
An adaptive integration scheme is used to counteract a tremendous integration effort,
that still is an essential criteria for the efficiency of the method if several large and dense
element matrices (e.g. stiffness, mass) are required, as they appear for higher polynomial
Ansatz functions. In the case of isotropic material properties the orthogonality of the
cell structure can be favorably exploited since the mapping of a global cell element to a
normalized hexahedral reference element is considerably simplified and allows a subcellbased
pre-computation of an element-independent stiffness and mass matrix except for
scaling factors representing material constants of each subcell. An optimal configuration
of polynomial degree and mesh density will be derived from several benchmark tests and
62

Contributed talks
Figure 3.1: Convergence plot for the first three circular frequencies of a cantilever beam
with circular cross-section and a moderate number of cells and p-elements,
respectively. A consistent mass distribution is applied. The numerical results
are compared to the analytical Bernoulli solution as an approximation to the
exact 3D solution as indicated by the curves levelling off at p > 3.
systematically analysed concerning accuracy, numerical stability and efficiency.
This contribution will give a brief overview about the basic idea of the Finite Cell
Method for dynamic problems including mass lumping strategies and the proper choice
of boundary conditions. A frequency analysis on cell and domain level is presented and
compared to results from classical h- and p-FEM computations to reveal the potentials
and limitations of the method for dynamic analyses. The proposed method is applied
to multi-material continuum problems.
References
[1] A. Düster, J. Parvizian, Z. Yang, E. Rank. “The finite cell method for three- dimensional
problems of solid mechanics”, Comput. Methods Appl. Mech. Engrg, v. 197, p. 3768-3782,
2008
[2] J. Parvizian, A. Düster, E. Rank. “Finite cell method – h- and p-extension for embedded
domain problems in solid mechanics”, Comput. Mech., v. 41, p. 121-133, 2007
[3] D. Schillinger, S. Kollmannsberger, R.-P. Mundani, E. Rank. “The Hierarchical B- Spline
Version of the Finite Cell Method for Geometrically Nonlinear Problems of Solid Mechanics”,
in Proc. IV ECCM, Paris, France, May 16-21, 2010
63

Contributed talks
Analysis of the Discrete Maximum Principle in the Mimetic
Finite Difference Method for Elliptic Problems
D. Svyatskiy
Applied Mathematics and Plasma Physics Group, Theoretical Division,
Los Alamos National Laboratory, Los Alamos, NM 87545, USA
dasvyat@lanl.gov
The Maximum Principle is one of the most important properties of solutions of partial
differential equations. Its numerical analog, the Discrete Maximum Principle (DMP), is
one of the properties that are very difficult to incorporate into numerical methods, especially
when the computational mesh contains distorted or degenerate cells or the problem
coefficients are highly heterogeneous and anisotropic. A violation of DMP may lead to
numerical instabilities such as oscillations and to non-physical solutions, for example,
when heat flows from a cold material to a hot one. Some physical quantities, like concentration
and temperature, are non-negative by their nature and their approximations
should be non-negative as well.
The family of the Mimetic Finite Difference (MFD) methods provides flexibility in the
choice of parameters which define a particular member of the family. For example, the
MFD discretization scheme for quadrilateral meshes depends on three parameters. The
correct choice of these parameters may guarantee that the resulting numerical scheme
satisfy the DMP principle. The analysis of this strategy is based on the properties of Mmatrices.
The monotonicity limits of MFD method are investigated in several practically
important cases including meshes generated using the Adaptive Mesh Refinement (AMR)
strategy.
64

List of participants
65

Jade Gesare Abuga jadeabuga@yahoo.com
Mathematics, University of Eastern Africa, Baraton
P.O.Box 2500, 30100, Eldoret, Kenya
Paola Antonietti paola.antonietti@polimi.it
MOX-Dipartimento di Matematica, Politecnico di Milano
Via Bonardi 9, 20133 Milano
Blanca Ayuso de Dios bayuso@crm.cat
Centre de Recerca Matematica (CRM), UAB Science Faculty
08193 Bellaterra Barcelona, Spain
Douglas Arnold arnold@ima.umn.edu
List of participants
Institut for Mathematics and its Applications, University of Minnesota
Minneapolis, MN 55455, USA
Ferdinando Auricchio auricchio@unipv.it
Dipartimento di Meccanica Strutturale, Università di Pavia
Via Ferrata 1, 27100 Pavia
Lourenco Beirão da Veiga lourenco.beirao@unimi.it
Dipartimento di Matematica “F. Enriques”, Università di Milano
Via Saldini 50, 20133 Milano, Italy
Timo Betcke t.betcke@reading.ac.uk
Department of Mathematics, University of Reading
Whiteknights, PO Box 220, Reading, RG6 6AX, UK
Michel Bercovier berco@cs.huji.ac.il
School of Computer Science and Engineering, Hebrew University of Jerusalem
E.J. Safra Campus, Givat-Ram Jerusalem 91904, Israel
Pavel Bochev pbboche@sandia.gov
Applied Mathematics and Applications, Sandia National Laboratories
P.O. Box 5800, MS 1320, Albuquerque, NM 87185-1320
Daniele Boffi daniele.boffi@unipv.it
Dipartimento di Matematica, Università di Pavia
Via Ferrata 1, 27100 Pavia
Luca Bonaventura luca.bonaventura@polimi.it
MOX - Dipartimento di Matematica, Politecnico di Milano
Via Bonardi 9, 20133 Milano
Francesca Bonizzoni francesca1.bonizzoni@mail.polimi.it
MOX, Dipartimento di Matematica “F. Brioschi”, Politecnico di Milano
Via Bonardi 9, edificio 14 “La Nave”, VI piano, 20133 Milano
67

List of participants
Andrea Bressan andrea.bressan@unipv.it
Dipartimento di Matematica, Università di Pavia
Via Ferrata 1, 27100 Pavia
Franco Brezzi brezzi@imati.cnr.it
Istituto di Matematica Applicata e Tecnologie Informatiche, C.N.R.
Via Ferrata 1, 27100 Pavia
Annalisa Buffa annalisa@imati.cnr.it
Istituto di Matematica Applicata e Tecnologie Informatiche, C.N.R.
Via Ferrata 1, 27100 Pavia
Andrea Cangiani andrea.cangiani@unimib.it
Dipartimento di Matematica e Applicazioni
Via Roberto Cozzi 53, Edificio U5, 20125 Milano
Fausto Cavalli fausto.cavalli@ing.unibs.it
Dipartimento di Matematica, Università di Brescia
Via Valotti 9, Brescia Italia
Zhiming Chen zmchen@lsec.cc.ac.cn
LSEC, Institute of Computational Mathematics, Chinese Academy of Sci-
ences
Beijing 100190, China
Claudia Chinosi claudia.chinosi@mfn.unipmn.it
Dipartimento di Scienze e Tecnologie Avanzate, Università del Piemonte
Orientale
viale T. Michel 11, Alessandria
Durkbin Cho durkbin@imati.cnr.it
Istituto di Matematica Applicata e Tecnologie Informatiche, C.N.R.
Via Ferrata 1, 27100 Pavia, Italy
Lagat Robert Cheruiyot lagat.robert@yahoo.co.uk
Mathematics and physics, University of Eastern Africa
Baraton, P.O Box 2500 Eldoret
Snorre Christiansen snorrec@math.uio.no
CMA and Dept. Math., University of Oslo
PO Box 1053 Blindern, NO-0316 Oslo, Norway
Fehmi Cirak fc286@eng.cam.ac.uk
Department of Engineering, University of Cambridge
Trumpington Street, Cambridge CB2 1PZ, U.K.
68

Elaine Cohen cohen@cs.utah.edu
School of Computing, University of Utah
List of participants
50 S. Central Campus Drive, 3190 MEB, Salt Lake City, UT, 84112, USA
Martin Costabel costabel@univ-rennes1.fr
IRMAR, Institut Mathématique, Université de Rennes 1
Campus de Beaulieu, 35042 Rennes Cedex, France
Catterina Dagnino catterina.dagnino@unito.it
Dipartimento di Matematica, Università di Torino
Via Carlo Alberto 10, 10124 Torino, Italia
Monique Dauge monique.dauge@univ-rennes1.fr
IRMAR, Université de Rennes 1
Campus de Beaulieu, 35042 Renne Cedex, France
Carlo de Falco carlo.defalco@polimi.it
Dipartimento di Matematica, Politecnico di Milano
P.zza Leonardo da Vinci 32
Leszek Demkowicz leszek@ices.utexas.edu
ICES, The University of Texas at Austin
ACE 6.326, 201 E.24th Street, Austin, TX 78712, USA
Alan Demlow demlow@ms.uky.edu
Department of Mathematics, University of Kentucky
715 Patterson Office Tower, Lexington, Kentucky 40506, USA
Tor Dokken tor.dokken@sintef.no
Department of Applied Mathematics, SINTEF ICT
Forskningsveien 1, Oslo
Régis Duvigneau Regis.Duvigneau@sophia.inria.fr
OPALE Project-Team, INRIA Sophia-Antipolis
2004 route des Lucioles, 06902 Sophia-Antipolis, France
John Evans evans@ices.utexas.edu
Institute for Computational Engineering and Sciences, University of Texas
at Austin
3517 North Hills Drive, #T301, Austin, TX, USA 78731
Richard Falk falk@math.rutgers.edu
Marco Favino favinom@usi.ch
Department of Mathematics, Rutgers University
110 Frelinghuysen Road, Piscataway, NJ 08854
Institute of Computational Science, University of Lugano
Via Giuseppe Buffi 13, Lugano, 6900
69

List of participants
Algiane Froehly algiane.froehly@inria.fr
INRIA
351,cours de la liberation,Bat A29 bis,33405 Talence Cedex, France
Gwenael Gabard gabard@soton.ac.uk
ISVR, University of Southampton
University Road, Southampton, SO17 1BJ, UK
Francesca Gardini francesca.gardini@unipv.it
Dipartimento di Matematica, Università di Pavia
Via Ferrata 1, 27100 Pavia, Italy
Lucia Gastaldi lucia.gastaldi@ing.unibs.it
Dipartimento di Matematica, Università di Brescia
Via Valotti 9, 25133 Brescia, Italy
Paolo Gatto gatto@ices.utexas.edu
ICES, University of Texas at Austin
ACE 4.102, 201 E 24th Street, Austin, TX 78712, U.S.A.
Loredana Gaudio loredana.gaudio@polimi.it
MOX - Dipartimento di Matematica “F. Brioschi”, Politecnico di Milano
Via Bonardi 9, 20133 Milano
Ivan Georgiev john@parallel.bas.bg
Institute of Mathematics and Informatics, Bulgarian Academy of Sciences
Acad. G. Bonchev St. 8, 1113 Sofia, Bulgaria
Irina Georgieva irina@math.bas.bg
Mathematical Modeling, Institute of Mathematics and Informatics, Bulgar-
ian Academy of Sciences
Bulgaria, Sofia 1113, ul. Acad. Georgi Bonchev 8
Marc Gerritsma M.I.Gerritsma@TUDelft.nl
Faculty of Aerspace Engineering, TU Delft
Kluyverweg 1, 2629 HS Delft, The Netherlands
Antti Hannukainen antti.hannukainen@tkk.fi
Department of Mathematics and Systems Analysis, Aalto University
P.O. Box 11100, FI-00076 Aalto, Finland
Christoph Heinrich heinrich@ma.tum.de
Centre for Mathematical Sciences, Technische Universität München
Fakultät für Mathematik (M2), Lehrstuhl für Numerische Mathematik, Boltz-
mannstraße 3 D, 85748 Garching bei München, Germany
70

Holger Heumann hheumann@math.ethz.ch
Seminar for Applied Mathematics, ETH Zürich
Rämistrasse 101, 8092 Zürich, Switzerland
Ralf Hiptmair hiptmair@sam.math.ethz.ch
SAM, D-MATH, ETH Zürich
Rämistrasse 101, CH 8092 Zürich
Anil Hirani hirani@cs.illinois.edu
List of participants
Department of Computer Science, University of Illinois at Urbana-Champaign
201 N. Goodwin Ave., Urbana, IL 61801, USA
Clemens Hofreither clemens.hofreither@dk-compmath.jku.at
DK Computational Mathematics, JKU University Linz
Altenbergerstr. 69, 4040 Linz, Austria
Akhlaq Husain akhlaq@iitk.ac.in
Mathematics, Indian Institute of Technology Kanpur
Department o Mathematics and Statistics, IIT Kanpur, Kanpur, Pin: 208016,
India
Bert Jüttler bert.juettler@jku.at
Institute of Applied Geometry, Johannes Kepler University
Altenberger Str. 69, 4040 Linz, Austria
Clement Jourdana clement@imati.cnr.it
Dipartimento di matematica, Università di Pavia
Via Ferrata 1, 27100 Pavia, Italy
Peter Knabner knabner@am.uni-erlangen.de
Department of Mathematics, Friedrich-Alexander Universität Erlangen-Nürn-
berg
Alexander Kolpakov algk@ngs.ru
Martensstrasse 3 D 91058 Erlangen
DAEIMI, Cassino University
Via Di Biasio, 43 03043 Cassino (FR), Italy
Jasper Kreeft j.j.kreeft@tudelft.nl
Aerospace Engineering - Aerodynamics, Delft University of Technology
Kluyverweg 1, 2629 HS Delft, The Netherlands
Mukesh Kumar mukesh@imati.cnr.it
Instituto di Matematica Applicata e Tecnologie Iinformatiche, CNR
Via Ferrata 1, 27100 Pavia
71

List of participants
Joachim Linn joachim.linn@itwm.fraunhofer.de
Mathematical Methods in Dynamics and Durability, Fraunhofer Institute
for Industrial Mathematics (ITWM)
Konstantin Lipnikov lipnikov@lanl.gov
Fraunhofer Platz 1 D, 67663 Kaiserslautern, Germany
Applied Mathematics and Plasma Physics, Los Alamos National Laboratory
MS B284, Los Alamos, NM 87545, USA
Carlo Lovadina carlo.lovadina@unipv.it
Dipartimento di Matematica, Università di Pavia
Via Ferrata 1, 27100 Pavia
Teemu Luostari teemu.luostari@uef.fi
Department of Physics and Mathematics, University of Eastern Finland
University of Eastern Finland, Department of Physics and Mathematics,
Kuopio Campus, P.O. Box 1627, FI-70211 Kuopio, Finland
Carla Manni manni@mat.uniroma2.it
Matematica, Universita di Roma “Tor Vergata”
Via della Ricerca Scientifica 1, 00133 Roma
Gianmarco Manzini marco.manzini@imati.cnr.it
IMATI, CNR
Via Ferrata 1, 27100 Pavia, Italia
Aurora Mihaela Marica marica@bcamath.org
Ikerbasque, Basque Center for Applied Mathematics
Bizkaia Technology Park, Building 500, 48160, Derio, Basque Country, Spain
Ilario Mazzieri ilario.mazzieri@mail.polimi.it
Politecnico di Milano
Via Bonardi 9, 20133 Milano
Azaiez Mejdi azaiez@ipb.fr
Mecanics, Institut Polytechnique de Bordeaux
Jens Markus Melenk melenk@tuwien.ac.at
Lab TREFLE, 16 Av Pey Berland 33607 Pessac Cedex
Institut für Analysis und Scientific Computing (E101), Technische Univer-
sität Wien
Wiedner Hauptstraße 8-10, A-1040 Wien, Austria
Andrea Moiola andrea.moiola@sam.math.ethz.ch
Seminar for Applied Mathematics, ETH Zürich
HG J 45, Rämistrasse 101, 8092 Zürich, CH
72

Peter Monk monk@math.udel.edu
List of participants
Department of Mathematical Sciences, University of Delaware
Newark, DE 19716, USA
Dang Manh Nguyen d.m.nguyen@mat.dtu.dk
Department of Mathematics, Technical University of Denmark
Matematiktorvet, Building 303S, Kgs. Lyngby, Denmark
Peter Nørtoft Nielsen p.n.nielsen@mat.dtu.dk
DTU Mathematics, Technical University of Denmark
Bygning 303S, DK-2800 Kgs. Lyngby, Denmark
Artur Palha a.palha@tudelft.nl
Aerodynamics Department, Faculty of Aerospace Engineering, TUDelft
Kluyverweg 2, 2629 HT Delft, The Netherlands
Asieh Parsania asieh.parsania@math.uzh.ch
Department of Mathematics, Zürich University
Winterthurerstrasse 190, CH-8057 Zürich, Switzerland
Luca Pavarino luca.pavarino@unimi.it
Department of Mathematics, Università di Milano
Via Saldini 50, 20133 Milano
Ilaria Perugia ilaria.perugia@unipv.it
Dipartimento di Matematica, Università di Pavia
Via Ferrata 1, 27100 Pavia
Jorg Peters jorg@cise.ufl.edu
CISE, University of Florida
Gainesville FL 32605
Joel Phillips joel.phillips@reading.ac.uk
Mathematics, Reading University
Whiteknights, PO Box 220, Reading RG6 6AX, UK
Paola Pietra pietra@imati.cnr.it
IMATI, CNR
Via Ferrata 1, 27100 Pavia
Valentina Poletti valentina.poletti@gmail.com
Computational Science Institute, University of Lugano
Via Giuseppe Buffi 13, Lugano, 6900
Alessio Quaglino quaglino@math.uni-goettingen.de
Numerical and Applied Maths, Göttingen University
Lotzestr. 16 Göttingen 37073 Germany
73

List of participants
Ahmed Ratnani ahmed.ratnani@math.unistra.fr
Applied mathematics, IRMA University of Strasbourg, INRIA Grand-Est
7 rue René-Descartes, 67084 Strasbourg Cedex, France
Alessandro Reali alessandro.reali@unipv.it
Structural Mechanics Department, University of Pavia
Via Ferrata 1, 27100, Pavia, Italy
Elisabetta Repossi elisabetta.repossi@tiscali.it
MOX- Dipartimento di Matematica, Politecnico di Milano
Via Bonardi, 9 20133 Milano, Italia
Judith Rivas judith.rivas@ehu.es
Martin Ruess ruess@tum.de
Mathematics, University of the Basque Country
Barrio Sarriena s/n, 48940 Leioa, Spain
Center for Simulation Technology in Engineering, Technische Universität
München
Arcisstraße 21, 80333 München
Gianni Sacchi gianni.sacchi@imati.cnr.it
IMATI, CNR
Via Ferrata 1, 27100 Pavia, Italy
Giancarlo Sangalli giancarlo.sangalli@unipv.it
Dipartimento di Matematica, Università di Pavia
Via Ferrata 1, 27100 Pavia
Marcus Sarkis msarkis@wpi.edu
Mathematical Sciences , Worcester Polytechnic Institute
100 Institute Road, Worcester, MA 01609, USA
Stefan Sauter stas@math.uzh.ch
Institut für Mathematik, Universität Zürich
Winterthurerstrasse 190, 8057 Zürich
Simone Scacchi simone.scacchi@unimi.it
Department of Mathematics, University of Milan
Via Saldini 50, 20133 Milano, Italia
Joachim Schöberl joachim.schoeberl@rwth-aachen.de
Institute for Analysis and Scientific Computing, Vienna University of Tech-
nology
Wiedner Hauptstrasse 8-10, Wien, Austria
74

Corina Simian corina.simian@math.uzh.ch
List of participants
Numerical Analyses, Institute for Mathematics, Zürich University
Winterthurerstrasse 190, CH-8057 Zürich
Daniil Svyatskiy dasvyat@lanl.gov
Theoretical Division, Los Alamos National Laboratory
Mail Stop B284, Los Alamos National Laboratory, Los Alamos, NM 87545,
U.S.A.
Pragnesh Thakkar pragnesh.thakkar@nirmauni.ac.in
Mathematics Scection, Institute of Diploma Studies, Nirma University
4,Varsha Apartment, Plot#184, Nehru Park, Opp.Vastrapur Lake, Vastra-
pur, Ahmedabad, Gujarat, India
Satyendra Tomar satyendra.tomar@ricam.oeaw.ac.at
RICAM, Austrian Academy of Sciences
Altenbergerstrasse 69, 4040 Linz, Austria
Rafael Vázquez vazquez@imati.cnr.it
Instituto di Matematica Applicata e Tecnologie Informatiche, CNR
Via Ferrata 1, 27100 Pavia, Italia
Alexander Veit alexander.veit@math.uzh.ch
Institut für Mathematik, University of Zürich
Winterthurerstrasse 190, CH-8057 Zürich
Marco Verani marco.verani@polimi.it
MOX-Department of Mathematics, Politecnico di Milano
Via Bonardi, 9 20133 Milano
Claudio Verdi claudio.verdi@unimi.it
Dipartimento di Matematica, Università di Milano
Via Saldini 50, 20133 Milano
Anh-Vu Vuong vuong@ma.tum.de
Centre for Mathematical Sciences, Technische Universität München
Fakultät für Mathematik, Lehrstuhl für Numerische Mathematik (M2), Boltz-
mannstraße 3 D, 85748 Garching, Germany
Mary F. Wheeler mfw@ices.utexas.edu
Center for Subsurface Modeling, The University of Texas at Austin
1 University Station C0200, Austin, Texas 78712
Ragnar Winther ragnar.winther@cma.uio.no
Centre of Mathematics for Applications (CMA), University of Oslo
CMA, P.O. Box 1053, Blindern, 0316 Oslo, Norway
75

List of participants
Gang Xu gxu@sophia.inria.fr
GALAAD Team, INRIA Sophia-antipolis
2004, route des Lucioles B.P. 93 F, 06902 Sophia Antipolis Cedex, France
Ivan Yotov yotov@math.pitt.edu
Department of Mathematics, University of Pittsburgh
301 Thackeray Hall, Pittsburgh, PA 15260, USA
76