Here - Hausdorff Center for Mathematics - Universität Bonn

Workshop
High-Order Numerical Approximation
for Partial Differential Equations
Hausdorff Center for Mathematics
University of Bonn
February 6–10, 2012
Organizers
Alexey Chernov (University of Bonn)
Christoph Schwab (ETH Zurich)

Program overview
Time Monday Tuesday Wednesday Thursday Friday
8:00 Registration
8:40 Opening
9:00
Stephan Sloan Ainsworth Demkowicz Buffa
10:00
10:30
Canuto Nobile Sherwin Houston
Beirão
da Veiga
Reali
11:00
Coffee break
11:30
12:00
Quarteroni Gittelson Schöberl Schötzau Sangalli
Bespalov
12:30
Lunch break
14:30
15:00
15:30
16:00
Dauge Xiu Wihler Düster
Excursions:
Arithmeum
Melenk Litvinenko Hesthaven
or
Maischak
16:30
Coffee break
17:00
–18:00
Botanic
Gardens
Coffee break
Hackbusch Rank Hiptmair
Free Time
Yosibash
Coffee break
Maday
20:00 Dinner
3

High-Order Numerical Approximation
for Partial Differential Equations
Over the last decades high-order numerical approximation for PDEs have become wellestablished
tools for the efficient, highly accurate numerical solution of (boundary)
integral equations and partial differential equations in computational science and engineering;
prominent examples are hp- and spectral element methods in application areas
such as computational fluid dynamics, computational climate modelling, to name but
a few. Their theoretical convergence analysis and adaptive hp- and spectral element
versions currently experience strong development. New application areas for spectral
methods are infinite dimensional Fokker-Planck equations, stochastic partial differential
equations, uncertainty quantification and isogeometric analysis. Here, analysis is
missing to a large extent and is subject of current research.
The aim of the Workshop is to bring together leading experts in analysis and implementation
of High-Order Methods for PDEs, thereby stimulating an intensive idea
exchange and fruitful interactions.
Workshop Venue
All talks of the workshop will take place at the Mathematics Center, Endenicher Allee 60,
in the Lipschitz Hall (Room 1.016), first floor (European counting). Please feel free to
use Rooms 1.007 and 1.008 as well as the blackboard in the Plücker Hall (Room 1.015)
for discussions. Please feel free to use the mathematics library in the ground floor. It
has a large collection of books and journals.
Lipschitz Hall
1.016
(lecture room)
Plücker Hall
1.015
(coffee
breaks)
Stairs up
Elevator
Stairs down
(Library /
Ground floor)
Discussion
Room
1.007
Main
Entrance
(1st Floor)
Discussion
Room
1.008
Endenicher Alle
4

Getting there
The most convenient way to get to the workshop venue from the hotel “My Poppelsdorf”
is walking (about 10-15 min). Alternatively, you might take a bus 631 from
the stop “Poppelsdorfer Platz” to the stop “Kaufmannstrasse” (4 stops, goes every 30
min.) or a taxi.
P
P
AS Bonn-
Endenich
U
ENDENICH
Röckumstr.
Flodelingsweg
Kapellenstr.
Immenburgstr.
WESTSTADT
Endenicher Str.
AS Bonn-
Poppelsdorf
A 565
Thomastr.
Endenicher Allee 60
Workshop Venue
Meckenheimer Allee 171
Botanic Gardens
Wallfahrtsweg 4
Hotel „My Poppelsdorf”
Sebastianstr.
NORDSTADT
Heerstr.
Wittelsbacherring
Haydnstr.
Bornheimer St r.
Kaufmannstr. Nußallee
Kreuzberg-
Humboldtstr.
Richard-Wagner-
Str.
Endenicher Allee
Carl-Troll-Str.
Wegelerstr.
Katzenburgweg
Alter
Friedhof
Sternenburgstr.
Clemens-August-Str.
Rabinstr.
Herwarthstr.
Colmantstr.
Beethovenstr.
Stadthaus
Poppelsdorfer
Schloss
Botanische
Gärten
H H
H
Baumschulallee
Prinz-
Beringstr.
weg
Kirschallee
Maxstr.
Th.-Mann-Str.
Meckenheimer Allee
H Hbf
DB
U
POPPELSDORF
Post
Quantiusstr.
Sternstr.
Gang
Poststr.
olfstr.
Poppelsdorfer Allee
str.
Maximilianstr.
H
H
H
SÜDSTADT
Reuterstr .
Maarflach
Kaiserpl.
Bonner Talweg
Bonngasse
Remigiusstr.
Sandkaule
Oxford-
Bonngasse 30
Restaurant „Im Stiefel”
str. B.-v.-Suttner-
H Platz
str.
Jagdweg Kurfürsten-
Venusbergstr.
Bismarckstr.
Königweg
Argelander-
Goeben-
Wenzelg.
Markt
Am Hof
Uni-
Hauptgebäude
Rathaus
U
Hofgarten
Kaiserstr.
Albert-
str.
str.
Wallfahrtsweg
Wesselstr.
A. d.
Schloßkirche
Str.
Belderberg
Regina-Pacis-
Münsterpl.
Friedrich-
Friedensplatz
Franziskanerstr.
Am
Stockenstr.
H
Weg
Am
Boeselagerhof
Giergasse
Opernhaus
Hofgarten
W eber-
Nassestr.
Konviktstr.
Schumannstr.
Adenauerallee
Lennéstr. 2
Arithmeum
Lennéstr.
Brassertufer
Koblenzer Tor
Kennedybrücke
H
An der
RHEIN
Erste
Fährg.
Rathenauufer
U
str.
Lessingstr. Arndtstr.
Elisabeth kirche
0 500 m
© Geographisches Institut der Universität Bonn
Lunch and coffee breaks
Coffee will be served during the coffee breaks in the Plücker Hall (Room 1.015), next to
the Lipschitz Hall where all talks will take place. For the lunch you might go to a place
of your choice. Many restaurants are located in the Clemens-August street (you should
have passed it on your way from the hotel “My Poppelsdorf” to the Workshop Venue)
or in the city center. Another option is the student restaurant (exit the Mathematics
Center, cross the street and turn to the right).
5

Internet access via WLAN
We offer a free Internet access via WLAN in the Workshop Venue. For this, you need
an access certificate. You should have received it via email in the beginning of January.
If you have difficulties with the Internet connection, please contact the registration desk
or the organizers.
Excursion on Wednesday
The Wednesday’s afternoon is free and we welcome you to join one of two excursions:
• Arithmeum (http://www.arithmeum.uni-bonn.de), a museum of historical
mechanical calculating machines, historical arithmetic books, etc.
• Green Houses of the Botanic Gardens of the University of Bonn
Both excursions start at 15:00 and will be accompanied by an English guide. You will
be expected at 15:00 at the entrance to Arithmeum (Lenné Str. 2) or Botanic Gardens
(Meckenheimer Allee 171). The locations are designated on the city plan.
The excursion in Arithmeum is free. A small fee (several Euros, depending on the
number of participants) applies for the excursion in the Botanic Gardens. We kindly
ask you to inform us during the registration, or by Monday evening the latest, which
excursion would you like to join.
Dinner on Thursday
On Thursday’s evening we welcome you to join us about 20:00 for the Workshop Dinner
at the “typical” restaurant Im Stiefel in the center of Bonn (Bonngasse 30). Unfortunately,
due to our budget rules we are not able to cover the costs for the Dinner.
Everybody will have to pay himself / herself. The food prices (excluding drinks) range
between 8 and 14 Euro.
We kindly ask you to inform us during the registration, or by Monday evening the
latest, if you wish to join us for the Workshop Dinner.
Acknowledgement
The organizers would like to thank the Hausdorff Center for Mathematics for the
generous financial support, which made this Workshop possible.
Special thanks go to Laura Siklossy and other members of the HIM and HCM Administration
& Support teams for the strong an continuous support in all administrative
issues. Many thanks to Gunder-Lily Sievert and her team for organizing the coffee
breaks. We also thank Anne Reinarz, Duong Pham and Claudio Bierig for helping us
with some practical issues before and during the Workshop.
6

Monday, February 6
Detailed program
9:00–9:55 Ernst P. Stephan
hp-adaptive DG-FEM for Parabolic Obstacle Problems
10:00–10:55 Claudio Canuto
Adaptive Fourier-Galerkin Methods
11:00–11:30 Coffee break
11:30–12:25 Alfio Quarteroni
Discontinuous approximation of elastodynamics equations
12:30–14:30 Lunch break
14:30–15:25 Monique Dauge
Weighted analytic regularity in polyhedra
15:30–16:25 Jens Markus Melenk
Stability and convergence of Galerkin discretizations of the Helmholtz equation
16:30–17:00 Coffee break
17:00–17:55 Wolfgang Hackbusch
Selfadaptive Compression in Tensor Calculus
Tuesday, February 7
9:00–9:55 Ian H. Sloan
PDE with random coefficients as a problem in high dimensional integration
10:00–10:55 Fabio Nobile
Stochastic Polynomial approximation of PDEs with random coefficients
11:00–11:30 Coffee break
11:30–12:00 Claude J. Gittelson
Potential and Limitations of Sparse Tensor Product Stochastic Galerkin Methods
12:00–12:30 Alex Bespalov
A priori error analysis of stochastic Galerkin mixed finite element methods
12:30–14:30 Lunch break
14:30–15:25 Dongbin Xiu
A Flexible Stochastic Collocation Algorithm on Arbitrary Nodes via Interpolation
15:30–16:00 Alexander Litvinenko
Computation of maximum norm and level sets in the canonical tensor format with
application to SPDE
16:00–16:30 Matthias Maischak
p-FEM and p-BEM – achieving high accuracy
16:30–17:00 Coffee break
17:00–17:55 Ernst Rank
To mesh or not to mesh: High order numerical simulation for complex structures
7

Wednesday, February 8
9:00–9:55 Mark Ainsworth
Bernstein-Bézier finite elements of arbitrary order and optimal assembly procedures
10:00–10:55 Spencer Sherwin
From h to p efficiently: balancing high or low order approximations with accuracy
11:00–11:30 Coffee break
11:30–12:25 Joachim Schöberl
Domain Decomposition Preconditioning for High Order Hybrid Discontinuous Galerkin
Methods on Tetrahedral Meshes
12:30–14:30 Lunch break
14:30–18:00 Excursions to “Arithmeum” or Botanic Gardens, free time
Thursday, February 9
9:00–9:55 Leszek Demkowicz
DPG method with optimal test functions. A progress report.
10:00–10:55 Paul Houston
Application of hp-Adaptive Discontinuous Galerkin Methods to Bifurcation Phenomena
in Pipe Flows
11:00–11:30 Coffee break
11:30–12:25 Dominik Schötzau
Exponential convergence of hp-version discontinuous Galerkin methods for elliptic problems
in polyhedral domains
12:30–14:30 Lunch break
14:30–15:25 Thomas P. Wihler
A Posteriori Error Analysis for Linear Parabolic PDE based on hp-discontinuous Galerkin
Time-Stepping
15:30–16:25 Jan S. Hesthaven
Certified Reduced Basis Methods for Integral Equations with Applications to Electromagnetics
16:30–17:00 Coffee break
17:00–17:55 Ralf Hiptmair
Convergence of hp-BEM for the electric field integral equation
8

Friday, February 10
9:00–9:55 Annalisa Buffa
On local refinement in isogeometric analysis
10:00–10:30 Lourenco Beirão da Veiga
Domain decomposition methods in Isogeometric analysis
10:30–11:00 Alessandro Reali
Isogeometric Collocation Methods for Elasticity
11:00–11:30 Coffee break
11:30–12:25 Giancarlo Sangalli
Isogeometric discretizations of the Stokes problem
12:30–14:30 Lunch break
14:30–15:00 Alexander Düster
Numerical homogenization procedures in solid mechanics using the finite cell method
15:00–15:30 Zohar Yosibash
Simulating the mechanical response of arteries by p-FEMs
15:30–16:00 Coffee break
16:00–16:55 Yvon Maday
A generalized empirical interpolation method based on moments and application
9

Monday, 9:00–9:55
hp-adaptive DG-FEM for Parabolic
Obstacle Problems
Ernst P. Stephan
stephan@ifam.uni-hannover.de
Insitut für Angewandte Mathematik (IfAM), Leibniz Universität Hannover (LUH),
Germany
Joint work with:
Lothar Banz (IfAM, LUH, Germany)
Parabolic obstacle problems find applications in the financial markets for pricing American
put options. We present a mixed method and an equivalent variational inequality
method, both based on an hp-interior penalty DG (IPDG) method combined with
an hp-time DG (TDG) method, to solve parabolic obstacle problems approximatively
[1]. The contact conditions are resolved by a biorthogonal Lagrange multiplier and
are component-wise decoupled. These decoupled contact conditions are equivalent to
finding the root of a non-linear complementary function. This non-linear problem can
in turn be solved efficiently by a semi-smooth Newton method. For the hp-adaptivity
a p-hierarchical error estimator in conjunction with a local analyticity estimate is employed,
where the latter decides weather an h or p-refinment should be carried out. For
the stationary problem, this leads to exponential convergence, and for the instationary
problem to greatly improved convergence rates.
Numerical experiments are given demonstrating the strengths and limitations of the
approaches.
References
[1] L. Banz hp-Finite Element and Boundary Element Methods for Elliptic, Elliptic
Stochastic, Parabolic and Hyperbolic Obstacle and Contact Probelems, PhD-Thesis
(submitted).
10

Monday, 10:00–10:55
Adaptive Fourier-Galerkin Methods
Claudio Canuto
claudio.canuto@polito.it
Politecnico di Torino
Joint work with:
R.H. Nochetto (University of Maryland),
M. Verani (Politecnico di Milano)
The design of adaptive spectral-element (or h-p) discretization algorithms for elliptic
self-adjoint problems relies on the dynamic interplay between two fundamental stages:
the refinement of the geometric decomposition into elements and the enrichment of the
local basis within an element. While the former stage is by now well understood in
terms of practical realization and optimality properties, less theoretical attention has
been devoted to the latter one.
With the aim of shedding light on this topic, we focus on what happens in a single element
of the decomposition. In order to reduce at a minimum the technical burdens, we
actually assume periodic boundary conditions on the d-dimensional box Ω = (0, 2π) d ,
in order to exploit the orthogonality properties of the Fourier basis. Thus, we consider
a fully adaptive Fourier method, in which at each iteration of the adaptive algorithm
the Galerkin solution is spanned by a dynamically selected, finite subset of the whole
set of Fourier basis functions. The active set is determined by looking at a fixed fraction
of largest (scaled) Fourier coefficients of the residual, according to the bulk-chasing (or
Dörfler marking) philosophy. The algorithm is proven to be convergent; in addition,
exploiting the concentration properties of the inverse of the elliptic operator represented
in the Fourier basis, one can show that the error reduction factor per iteration
tends to 0 as the bulk-chasing fraction tends to 1.
After convergence has been established, one is faced with the issue of optimality. This
leads to the comparison of the adaptive Galerkin solution spanned by, say, N Fourier
modes, with the best N-term approximation of the exact solution. Consequently, we
are led to introduce suitable sparsity classes of periodic H 1 -functions for which the best
N-term approximation error fulfils a prescribed decay as N tends to infinity. These
classes can also be characterized in terms of behavior of the rearranged sequence of the
(normalized) Fourier coefficients of the functions.
If the best N-term approximation error of the exact solution decays at an algebraic
rate (this occurs if the solution belongs to a certain “oblique” scale of Besov spaces of
periodic functions, corresponding to a finite regularity), then the arguments developed
by Cohen, Dahmen and DeVore and by Stevenson et al in the framework of wavelet
bases apply to our situation as well, and one can establish the optimality of the approximation
without coarsening. The crucial ingredients in the analysis are the minimality
11

property of the active set of degrees of freedom determined by bulk-chasing, and a
geometric-series argument (essentially, the estimated number of degrees of freedom
added at each iteration is comparable to the total number of degrees of freedom added
in all previous iterations).
On the other hand, the case of a solution having (local) infinite-order or analytical
regularity is quite significant in dealing with spectral (spectral-element) methods. For
the Navier-Stokes equations, regularity results in Gevrey classes have been first established
by Foias and Temam; in these classes, the best N-term approximation error of
a function decays precisely at a sub-exponential or exponential rate. Then, the analysis
of an adaptive strategy becomes much more delicate, as the previous arguments
fail to apply. For instance, even for a linear operator with analytic ceofficients, it is
shown that the Gevrey sparsity class of the residual is different from (actually, worse
than) the one of the exact solution. Therefore, we devise and analyse a new version
of the adptive algorithm considered before: since the choice of the next set of active
degrees of freedom is made on the basis of the sparsity of the residual, we incorporate
a coarsening step in order to get close to the sparsity pattern of the solution.
The extension of some of the previous results to the case of non-periodic (Legendre)
expansions will also be considered.
12

Monday, 11:30–12:25
Discontinuous approximation
of elastodynamics equations
Alfio Quarteroni
alfio.quarteroni@epfl.ch
MATHICSE, EPFL, Lausanne and MOX, Politecnico di Milano, Milan
Joint work with:
P.Antonietti, I.Mazzieri and F.Rapetti
The possibility of inferring the physical parameter distribution of the Earth’s substratum,
from information provided by elastic wave propagations, has increased the interest
towards computational seismology. Recent developments have been focused on spectral
element methods. The reason relies on their flexibility in handling complex geometries,
retaining the spatial exponential convergence for locally smooth solutions and a natural
high level of parallelism. In this talk, we consider a Discontinuous Galerkin spectral
element method (DGSEM) as well as discontinuous mortar methods (DMORTAR) to
simulate seismic wave propagations in three dimensional heterogeneous media. The
main advantage with respect to conforming discretizations as those based on Spectral
Element Method is that DG and Mortar discretizations can accommodate discontinuities,
not only in the parameters, but also in the wavefield, while preserving the energy.
The domain of interest Ω is assumed to be union of polygonal substructures Ω i . We
allow this substructure decomposition to be geometrically non-conforming. Inside each
substructure Ω i , a conforming high order finite element space associated to a partition
T hi (Ω i ) is introduced. We allow the use of different polynomial approximation degrees
within different substructures. Applications to simulate benchmark problems as well
as realistic seismic wave propagation processes are presented.
13

Monday, 14:30–15:25
Weighted analytic regularity in polyhedra
Monique Dauge
Monique.Dauge@univ-rennes1.fr
IRMAR, CNRS and Université de Rennes 1, FRANCE
Joint work with:
Martin Costabel (IRMAR, Université de Rennes 1, FRANCE),
Serge Nicaise (LAMAV, Université de Valenciennes, FRANCE)
Elliptic boundary value problems with analytic data (coefficients, domain and right
hand sides) are analytic up to the boundary [5]. Such a regularity allows exponential
convergence of the p-version of the finite element method. This result does not hold in
the same form in domains with edges and corners.
In ca. 1988 Babuška and Guo started a program relying on three ideas
1. The p-version should be replaced by the hp-version of finite elements,
2. The hp-version is exponentially converging if solutions belong to certain weighted
analytic classes (which they call countably normed spaces),
3. The solutions to standard elliptic boundary problems belong to such analytic
classes.
They proved results covering these three points for bi-dimensional problems (Laplace,
Lamé). They started the investigation of three-dimensional problems in [2, 3, 4]. But
three-dimensional problems have a level of complexity higher than 2D, for two reasons
a) The hp-version requires anisotropic refinement along edges to prevent a blowing
up of the number of degrees of freedom. Exponential convergence with such discretization
will be obtain only if solutions belong to anisotropic analytic weighted
spaces.
b) The edge behavior combine in a non-trivial way at each corner.
In our paper [1], we prove anisotropic weighted regularity for a full class of coercive
variational problems, homogeneous with constant coefficients. This completes for the
first time point 3. of the Babuška-Guo program in polyhedra. In this talk, I present the
results of this paper. The case of Laplace-Dirichlet and Laplace-Neumann problems will
serve as threads along our way in the various possible combinations of weights in twoand
three-dimensional domains, and the various corresponding statements. General
statements will be provided as well.
14

References
[1] M. Costabel, M. Dauge, and S. Nicaise, Analytic Regularity for Linear Elliptic
Systems in Polygons and Polyhedra, Math. Models Methods Appl. Sci., 22 (2012),
No. 8.
arXiv: http://fr.arxiv.org/abs/1112.4263.
[2] B. Guo. The h-p version of the finite element method for solving boundary value
problems in polyhedral domains. In M. Costabel, M. Dauge, S. Nicaise, editors,
Boundary value problems and integral equations in nonsmooth domains (Luminy,
1993), pages 101–120. Dekker, New York 1995.
[3] B. Guo, I. Babuška. Regularity of the solutions for elliptic problems on nonsmooth
domains in R 3 . I. Countably normed spaces on polyhedral domains. Proc. Roy.
Soc. Edinburgh Sect. A, 127 (1997), No. 1, 77–126.
[4] B. Guo, I. Babuška. Regularity of the solutions for elliptic problems on nonsmooth
domains in R 3 . II. Regularity in neighbourhoods of edges. Proc. Roy. Soc. Edinburgh
Sect. A, 127 (1997), No. 3, 517–545.
[5] C. B. Morrey, Jr., L. Nirenberg. On the analyticity of the solutions of linear elliptic
systems of partial differential equations. Comm. Pure Appl. Math. 10 (1957), 271–
290.
15

Monday, 15:30–16:25
Stability and convergence of Galerkin
discretizations of the Helmholtz equation
J.M. Melenk
melenk@tuwien.ac.at
Vienna University of Technology
Joint work with:
Stefan Sauter (University of Zurich),
Maike Löhndorf (Kapsch TrafficCom)
We consider boundary value problems for the Helmholtz equation at large wave numbers
k. In order to understand how the wave number k affects the convergence properties
of discretizations of such problems, we develop a regularity theory for the Helmholtz
equation that is explicit in k. At the heart of our analysis is the decomposition of solutions
into two components: the first component is an analytic, but highly oscillatory
function and the second one has finite regularity but features wavenumber-independent
bounds.
This new understanding of the solution structure opens the door to the analysis of
discretizations of the Helmholtz equation that are explicit in their dependence on the
wavenumber k. As a first example, we show for a conforming high order finite element
method that quasi-optimality is guaranteed if (a) the approximation order p is selected
as p = O(log k) and (b) the mesh size h is such that kh/p is small.
As a second example, we consider combined field boundary integral equation arising
in acoustic scattering. Also for this example, the same scale resolution conditions as
in the high order finite element case suffice to ensure quasi-optimality of the Galekrin
discretization.
16

Monday, 17:00–17:55
Selfadaptive Compression in Tensor Calculus
Wolfgang Hackbusch
wh@mis.mpg.de
Max-Planck-Institut Mathematik in den Naturwissenschaften
Inselstr. 22, D-04103 Leipzig
The principle behind high-order methods is the wish to obtain high accuracy by a
small number of degrees of freedom. This leads, e.g., to the hp-method, where still
the question arises how the step sizes and polynomial degrees are to be chosen optimally.
A further question is whether polynomials should be replaced by another class
of functions. A similar situation occurs for wavelet approaches.
The tensor approaches for functions in product domains (e.g, in R d ) start usually with a
regular grid as known from old-fashioned difference methods. This seems to contradict
the modern approaches. However, the algorithms in tensor calculus are essentially
based on compression principle. In the best case, the total data size n d (n grid points
in each of the d directions) can be compressed to O(d · log n · log(1/ε)) in order to
obtain an approximation of accuracy ε. It turns out that the obtainable compression
is at least as good as obtainable by hp-methods (or other high-order) methods. The
essential difference is that compression methods in tensor calculus find the suitable
distributions of h and p automatically and even chooses the suitable class of functions
in a selfadaptive way.
References
[1] Hackbusch, Wolfgang: Convolution of hp-functions on locally refined grids. IMA J.
Numer. Anal. 29, 960–985 (2009)
[2] Hackbusch, Wolfgang: Tensor spaces and numerical tensor calculus. SCM 42,
Springer Berlin (2012)
[3] Khoromskij, Boris: O(d log N)-quantics approximation of N − d tensors in highdimensional
numerical modeling. Constr. Approx. (2011). To appear
[4] Oseledets, Ivan V.: Approximation of matrices using tensor decomposition. SIAM
J. Matrix Anal. Appl. 31, 2130–2145 (2010)
17

Tuesday, 9:00–9:55
PDE with random coefficients as a problem
in high dimensional integration
Ian H Sloan
i.sloan@unsw.edu.au
University of New South Wales
Joint work with:
Frances, Y Kuo (University of New South Wales),
Christoph Schwab (ETH Zurich)
This talk is concerned with the use of quasi-Monte Carlo methods combined with finiteelement
methods to handle an elliptic PDE with a random field as a coefficient. A number
of groups have considered such problems (under headings such as polynomial chaos,
stochastic Galerkin and stochastic collocation) by reformulating them as deterministic
problems in a high dimensional parameter space, where the dimensionality comes from
the number of random variables needed to characterise the random field. In a recent
paper we have treated a problem of this kind as one of infinite-dimensional integration
- where integration arises because a multi-variable expected value is a multidimensional
integral - together with finite- element approximation in the physical space. We use
recent developments in the theory and practice of quasi-Monte Carlo integration rules
in weighted Hilbert spaces, through which rules with optimal properties can be constructed
once the weights are known. The novel feature of this work is that for the first
time we are able to design weights for the weighted Hilbert space that achieve what is
believed to be the best possible rate of convergence, under conditions on the random
field that are exactly the same as in a recent paper by Cohen, DeVore and Schwab on
best N-term approximation for the same problem.
References
[1] A. Cohen, R. De Vore and C. Schwab Convergence rates of best N -term apporximations
for a class of elliptic sPDEs, Foundations of Computational Mathematics,
10 (2010), 615–646.
[2] F. Y. Kuo, C. Schwab and I.H. Sloan Quasi-Monte Carlo finite element methods for
a class of elliptic partial differential equations with random coefficients, submitted,
2011.
18

Tuesday, 10:00–10:55
Stochastic Polynomial approximation of PDEs
CSQI MATHICSE,
with random coefficients
Fabio Nobile
fabio.nobile@epfl.ch
École Polytechnique Fédérale de Lausanne, Station 8, CH-1015
Lausanne, Switzerland
Joint work with:
L. Tamellini (MOX, Department of Mathematics, Politecnico di Milano, Italy),
J. Beck, M. Motamed, R. Tempone (Applied Mathematics and Computational
Science King Abdullah University of Science and Technology (KAUST), Kingdom of
Saudi Arabia)
We consider the problem of numerically approximating statistical moments of the solution
of a partial differential equation (PDE), whose input data (coefficients, forcing
terms, boundary conditions, geometry, etc.) are uncertain and described by a finite
or countably infinite number of random variables. This situation includes the case of
infinite dimensional random fields suitably expanded in e.g Karhunen-Loève or Fourier
expansions.
We focus on polynomial chaos approximations of the solution with respect to the
underlying random variables and review common techniques to practically compute
such polynomial approximation by Galerkin projection, Collocation on sparse grids or
regression methods from random evaluations.
We discuss in particular the proper choice of the polynomial space both for linear elliptic
PDEs with random diffusion coefficient and second order hyperbolic equations with
random piecewise constant wave speed. Numerical results showing the effectiveness
and limitations of the approaches will be presented as well.
References
[1] G. Migliorati, F. Nobile, E. von Schwerin, and R. Tempone, “Analysis of the discrete
L 2 projection on polynomial spaces with random evaluations,” MOX-Report 46-
2011, Department of Mathematics, Politecnico di Milano, Italy, 2011. submitted.
[2] M. Motamed, F. Nobile, and R. Tempone, “A stochastic collocation method for
the second order wave equation with a discontinuous random speed,” MOX-Report
36-2011, Department of Mathematics, Politecnico di Milano, Italy, 2011. submitted.
19

[3] J. Bäck, F. Nobile, L. Tamellini, and R. Tempone, “On the optimal polynomial
approximation of stochastic PDEs by galerkin and collocation methods,” MOX-
Report 23-2011, Department of Mathematics, Politecnico di Milano, Italy, 2011.
accepted for publication on M3AS.
[4] I. Babuška, F. Nobile, and R. Tempone, “A stochastic collocation method for elliptic
partial differential equations with random input data,” SIAM Review, vol. 52,
pp. 317–355, June 2010.
20

Tuesday, 11:30–12:00
Potential and Limitations of Sparse Tensor Product
Stochastic Galerkin Methods
Claude J. Gittelson
cgittels@purdue.edu
Purdue University
Sparse tensor product constructions are known to be more efficient than their full
tensor product counterparts in many applications. In case of low regularity, spectral
approximations for random differential equations form an exception to this rule.
We consider an elliptic PDE with a random diffusion coefficient, which we assume to
be expanded in a series. The solution to such an equation depends on the coefficients
in this series in addition to the spatial variables. Spectral discretizations typically use
tensorized polynomials to represent the parameter-dependence of the solution. Each
coefficient of the solution with respect to such a polynomial basis is a function of the
spatial variable, and can be approximated by finite elements.
Since the importance of the polynomial basis functions can vary greatly, it is natural
to expect a gain in efficiency if the coefficients are approximated in different finite element
spaces. We show under what conditions such a multilevel construction reaches a
higher convergence rate than approximations employing just a single spatial discretization.
Sparse tensor products, which impose additional structure on the multilevel
approximation, can always essentially attain the optimal convergence rate.
References
[1] C.J. Gittelson, Convergence rates of multilevel and sparse tensor approximations
for a random elliptic PDE, Submitted.
21

Tuesday, 12:00–12:30
A priori error analysis of stochastic Galerkin
mixed finite element methods
Alex Bespalov
alexey.bespalov@manchester.ac.uk
School of Mathematics, University of Manchester, Oxford Road, Manchester,
M13 9PL, United Kingdom
Joint work with:
Catherine Powell and David Silvester
(School of Mathematics, University of Manchester, United Kingdom)
Stochastic Galerkin methods are becoming increasingly popular for solving PDE problems
with random data (i.e., coefficients, sources, boundary conditions, geometry etc.).
These methods combine conventional Galerkin finite elements on the (spatial) computational
domain with spectral approximations in the stochastic variables. Whilst there
exists a large body of research work on numerical approximation of primal formulations
for elliptic PDEs with random data, the case of mixed variational problems is not so
well developed.
In this talk we will address the issues involved in the error analysis of stochastic
Galerkin approximations to the solution of a first order system of PDEs with random
coefficients. First, we approximate the random input by using spectral expansions in M
random variables and transform the variational saddle point problem to a parametric
deterministic one. Approximations to the latter problem are then constructed by combining
mixed hp finite elements on the computational domain with M-variate tensor
product polynomials of order k = (k 1 , k 2 , . . . , k M ). We will discuss the inf-sup stability
and well-posedness of the continuous and finite-dimensional problems, the regularity
of solutions with respect to the M parameters describing the random coefficients, and
a priori error bounds for stochastic Galerkin approximations in terms of discretisation
parameters M, h, p, and k.
This work is supported by the EPSRC (UK) under grant no. EP/H021205/1.
22

Tuesday, 14:30–15:25
A Flexible Stochastic Collocation Algorithm
on Arbitrary Nodes via Interpolation
Dongbin Xiu
dxiu@purdue.edu
Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA
Joint work with:
Akil Narayan, Purdue University
Stochastic collocation method have become the dominating methods for uncertainty
quantification and stochastic computing of large and complex systems. Though the
idea has been explored in the past, its popularity is largely due to the recent advance
of employing high-order nodes such as sparse grids. These nodes allow one to conduct
UQ simulations with high accuracy and efficiency.
The critical issue is, without any doubt, the standing challenge of “curse-of-dimensionality”.
For practical systems with large number of random inputs, the number of nodes for
stochastic collocation method can grow fast and render the method computationally
prohibitive. Such kind of growth is especially severe when the nodal construction is
structured, e.g., tensor grids, sparse grids, etc. One way to alleviate the difficulty is to
employ adaptive approach, where the nodes are added only in the region that is needed.
To this end, it is highly desirable to design stochastic collocation methods that work
with arbitrary number of nodes on arbitrary locations. Another strong motivation is
the practical restriction one may face. In many cases one can not conduct simulations
at the desired nodes.
In this work we present an algorithm that allows one to construct high-order polynomial
responses based on stochastic collocation on arbitrary nodes. The method is based
on constructing a “correct” polynomial space so that multi-dimensional polynomial interpolation
can be constructed for any data. We present its rigorous mathematical
framework, its practical implementation details, and its applications in high dimensions.
23

Tuesday, 15:30–16:00
Computation of maximum norm and level sets
in the canonical tensor format with application
to SPDE
Alexander Litvinenko
wire@tu-bs.de
Technische Universität Braunschweig, Hans-Sommerstr. 65, 38106 Braunschweig
Joint work with:
M. Espig, W. Hackbusch (MPI for Mathematics in the Sciences, Leipzig, Germany),
H. G. Matthies, E. Zander (Technische Universität Braunschweig)
We introduce new methods for the analysis of high dimensional data in tensor formats,
where the underling data come from the stochastic elliptic boundary value problem.
After discretisation of the deterministic operator as well as the presented random fields
via KLE and PCE, the obtained high dimensional operator can be approximated via
sums of elementary tensors. This tensors representation can be effectively used for
computing different values of interest, such as maximum norm, level sets and cumulative
distribution function. The basic concept of the data analysis in high dimensions is
discussed on tensors represented in the canonical format, however the approach can be
easily used in other tensor formats. As an intermediate step we describe efficient iterative
algorithms for computing the characteristic and sign functions as well as pointwise
inverse in the canonical tensor format. Since during majority of algebraic operations
as well as during iteration steps the representation rank grows up, we use lower-rank
approximation and inexact recursive iteration schemes.
References
[1] M. Espig, W. Hackbusch, A. Litvinenko, H. G. Matthies and E. Zander, Efficient
Analysis of High Dimensional Data in Tensor Formats, Technische Universität
Braunschweig, preprint 11-2011, http://www.digibib.tu-bs.de/?docid=00041268
24

Tuesday, 16:00–16:30
p-FEM and p-BEM – achieving high accuracy
Matthias Maischak
matthias.maischak@brunel.ac.uk
Brunel University, Uxbridge UB8 3PH, UK
It is well known, that we can achieve exponential fast convergence by approximating
the solution of a pde using either the p-version fem or bem if then solution is smooth,
or using the hp-version with geometrical refined mesh if the solution has singularities
[2, 3]. Achieving exponential fast convergence asymptotically proves to be surprisingly
difficult, due to requirements on memory, solution time and the necessary numerical
precision.
In this presentation we investigate changes necessary to an existing fem/bem code,
software tools for converting the code to a different data type using a multi-precision
package [1] and discuss implementation issues which arise from considerable more time
and memory consuming arithmetic operations.
We will compare standard double precision (hardware supported), quad precision (compiler
supported) and higher precision (additional Software package [1]). We will present
examples achieving errors with 1e-30 or less using p- and hp fem-bem methods in 1,2
and 3 dimensions.
References
[1] David H. Bailey, Yozo Hida, Xiaoye S. Li, and Brandon Thompson, ARPREC: An
Arbitrary Precision Computation Package, Technical Report LBNL-53651, 2002
[2] B. Guo, and I. Babuška, The h − p version of the finite element method, part 1:
The basic approximation results, Computational Mechanics, 1 (1986), 21–41.
[3] N. Heuer, M. Maischak, and E.P. Stephan, Exponential convergence of the hpversion
for the boundary element method on open surfaces, Numerische Mathematik,
83 (1999), 641–666.
25

Tuesday, 17:00–17:55
To mesh or not to mesh: High order numerical
simulation for complex structures
Ernst Rank
rank@bv.tum.de
Chair for Computation in Engineering, Technische Universität München, Arcisstrasse
21, 80333 München, Germany
Joint work with:
Stefan Kollmannsberger, Dominik Schillinger, Christian Sorger (Chair for
Computation in Engineering, Technische Universität München, Arcisstrasse 21, 80333
München, Germany)
The paper will give an overview over some recent work on high order finite element
meshing for thin-walled structures as well as on a high order variant of fictitious domain
methods. The basis for our research is the p-version of the finite element method.
It allows for a very accurate computation of thin-walled structures using solid models
and is not depending on dimensional reduction like in shell or plate models. To
take full advantage of this method it is yet necessary to develop special mesh generating
techniques which discretize the exact 3-dimensional curved geometry and not a
facetted mid-surface of the structure. In the second part of the presentation the Finite
Cell Method (FCM), an extension of high order finite element methods to a fictitious
domain approach will be discussed. It completely avoids meshing, yet converges exponentially
in energy norm for smooth problems. This approach turns out to be very
efficient and accurate and can even be used for a real-time simulation of complex solid
structures. Very recent results on an extension of the Finite Cell Method by an adaptive,
hierarchic refinement allows a tight connection between geometric modeling and
numerical analysis for thin-walled as well as for massive structures.
26

Wednesday, 9:00–9:55
Bernstein-Bézier finite elements of arbitrary order
and optimal assembly procedures
Mark Ainsworth
m.ainsworth@strath.ac.uk
University of Strathclyde
Joint work with:
G. Andriamaro and O. Davydov
Algorithms are presented that enable the element matrices for the standard finite element
space, consisting of continuous piecewise polynomials of degree n on simplicial
elements in R d , to be computed in optimal complexity O(n 2 d). The algorithms (i) take
account of numerical quadrature; (ii) are applicable to non-linear problems; and, (iii)
do not rely on pre-computed arrays containing values of one-dimensional basis functions
at quadrature points (although these can be used if desired). The elements are
based on Bernstein-Bézier polynomials and are the first to achieve optimal complexity
for the standard finite element spaces on simplicial elements.
27

Wednesday, 10:00–10:55
From h to p efficiently: balancing high or low order
approximations with accuracy
Spencer Sherwin
s.sherwin@imperial.ac.uk
Department of Aeronautics, Imperial College London, UK
s.sherwin@imperial.ac.uk
When approximating a PDE problem a reasonable question to ask is what is the ideal
discretisation order for a desired computational error? Although we typically understand
the answer in rather general terms there are comparatively little quantitive
results? The issues behind this question are partly captured in the figure above where
we observe the wall shear stress pattern around two small branching vessels in and
anatomically realistic mode of part of the descending aorta. In the left image we observe
a linear finite element approximation on a fine mesh and in the right image a
coarse mesh approximation using a 6th order polynomial expansion is applied. The
high order discretisation is much smoother but this accuracy typically is associated
with a higher computational cost. Although the low order approximation looks more
pixelated it does however capture all the salient features of the wall shear stress patterns.
The spectral element and hp finite element methods methods can be considered as
bridging the gap between the – traditionally low order – finite element method on one
side and spectral methods on the other side. Consequently, a major challenge which
arises in implementing the spectral/hp element methods is to design algorithms that
perform efficiently for both low- and high-order discretisations, as well as discretisations
in the intermediate regime.
In this presentation, we explain how the judicious use of different implementation
strategies can be employed to achieve high efficiency across a range of polynomial
28

orders [1, 2]. Further, based upon this efficient implementation, we analyse which
spectral/hp discretisation minimises the computational cost to solve elliptic steady
and unsteady model problems up to a predefined level of accuracy. Finally we discuss
how the lessons learned from this study can be encapulated into a more generic library
implementation.
References
[1] P.E.J. Vos, S.J. Sherwin and R.M. Kirby, From h to p efficiently: Implementing
finite and spectral/hp element discretisations to achieve optimal performance at
low and high order approximations, Journal of Computational Physics 229, (2010),
5161-5181
[2] P.E.J. Vos, S.J. Sherwin and R.M. Kirby, C.D. Cantwell, S.J. Sherwin, R.M. Kirby
and P.H.J. Kelly, From h to p efficiently: Strategy selection for operator evaluation
on hexahedral and tetrahedral elements, Computers and Fluids, to appear, 2011
29

Wednesday, 11:30–12:25
Domain Decomposition Preconditioning for High
Order Hybrid Discontinuous Galerkin Methods
on Tetrahedral Meshes
Joachim Schöberl
joachim.schoeberl@tuwien.ac.at
Vienna University of Technology
Hybrid discontinuos Galerkin methods are popular discretization methods in applications
from fluid dynamics and many others. Often large scale linear systems arising
from elliptic operators have to be solved. We show that standard p-version domain
decomposition techniques can be applied, but we have to develop new technical tools
to prove poly-logarithmic condition number estimates, in particular on tetrahedral
meshes.
30

Thursday, 9:00–9:55
DPG method with optimal test functions.
A progress report.
Leszek Demkowicz
leszek@ices.utexas.edu
University of Texas at Austin
I will present a progress report on the Discontinuous Petrov-Galerkin method proposed
by Jay Gopalakrishan and myself [1, 2]. The main idea of the method is to employ
(approximate) optimal test functions that are computed on the fly at the element
level. If the error in approximating the optimal test functions is negligible, the method
AUTOMATICALLY guarantees the discrete stability, provided the continuous problem
is well posed. And this holds for ANY linear problem. The result is shocking until one
realizes that we are working with an unconventional least squares method. The twist
lies in the fact that the residual lives in a dual space and it is computed using dual
norms.
The method turns out to be especially suited for singular perturbation problems where
one strives not only for stability but also for ROBUSTNESS, i.e. a stability UNIFORM
with respect to the perturbation parameter. Critical to the construction of a robust
version of the DPG method is the choice of the test norm that should imply the uniform
stability.
I will report on recent results for two important model problems: convection-dominated
diffusion and linear acoustics equations [3, 4, 5]. I will attempt to outline a general
strategy for selecting the optimal test norm and will focus on the task of approximating
the optimal test functions and effects of such an approximation [6].
References
[1] L. Demkowicz and J. Gopalakrishnan. A Class of Discontinuous Petrov-Galerkin
Methods. Part II: Optimal Test Functions. Numer. Meth. Part. D. E., 27, 70-105,
2011.
[2] L. Demkowicz, J. Gopalakrishnan and A. Niemi, A Class of Discontinuous Petrov-
Galerkin Methods. Part III: Adaptivity, ICES Report 2010-01, App. Num Math.,
in print.
[3] L. Demkowicz, J. Gopalakrishnan, I. Muga, and J. Zitelli. Wavenumber Explicit
Analysis for a DPG Method for the Multidimensional Helmholtz Equation”, ICES
Report 2011-24, CMAME, in print.
31

[4] L. Demkowicz, N. Heuer, Robust DPG Method for Convection-Dominated Diffusion
Problems. ICES Report 2011-33, submitted to SIAM J. Num. Anal.
[5] L. Demkowicz, J. Gopalakrishnan, I. and Muga, D. Pardo, A Pollution Free DPG
Method for Multidimensional Helmholtz Equation, in preparation.
[6] J. Gopalakrishnan and W. Qiu. An Analysis of the Practical DPG Method. IMA
Report 2011-2374.
32

Thursday, 10:00–10:55
Application of hp–Adaptive Discontinuous Galerkin
Methods to Bifurcation Phenomena in Pipe Flows
Paul Houston
Paul.Houston@nottingham.ac.uk
School of Mathematical Sciences, University of Nottingham,
Nottingham NG7 2RD, UK
In this talk we consider the a posteriori error estimation and hp–adaptive mesh refinement
of discontinuous Galerkin finite element approximations of the bifurcation
problem associated with the steady incompressible Navier–Stokes equations. Particular
attention is given to the reliable error estimation of the critical Reynolds number
at which a steady pitchfork bifurcation occurs when the underlying physical system
possesses rotational and reflectional or O(2) symmetry. Here, computable a posteriori
error bounds are derived based on employing the generalization of the standard
Dual Weighted Residual approach, originally developed for the estimation of target
functionals of the solution, to bifurcation problems; see [1, 2, 3] for details. Numerical
experiments highlighting the practical performance of the proposed a posteriori error
indicator on h– and hp–adaptively refined computational meshes are presented. Here,
particular attention is devoted to the problem of flow through a cylindrical pipe with
a sudden expansion, which represents a notoriously difficult computational problem.
This research has been carried out in collaboration with Andrew Cliffe and Edward Hall
(University of Nottingham), Tom Mullin and James Seddon (University of Manchester),
and Eric Phipps and Andy Salinger (Sandia National Laboratories).
References
[1] K.A. Cliffe, E. Hall, P. Houston, E.T. Phipps, and A.G. Salinger. Adaptivity and
A Posteriori Error Control for Bifurcation Problems I: The Bratu Problem. Communications
in Computational Physics 8(4):845-865, 2010.
[2] K.A. Cliffe, E. Hall, P. Houston, E.T. Phipps, and A.G. Salinger. Adaptivity and A
Posteriori Error Control for Bifurcation Problems II: Incompressible fluid flow in
Open Systems with Z 2 Symmetry. Journal of Scientific Computing 47(3):389-418,
2011.
[3] K.A. Cliffe, E. Hall, P. Houston, E.T. Phipps, and A.G. Salinger. Adaptivity and
A Posteriori Error Control for Bifurcation Problems III: Incompressible fluid flow
in Open Systems with O(2) Symmetry. Journal of Scientific Computing (in press).
33

Thursday, 11:30–12:25
Exponential convergence of hp-version
discontinuous Galerkin methods for elliptic
problems in polyhedral domains
Dominik Schötzau
schoetzau@math.ubc.ca
Mathematics Department, University of British Columbia, Vancouver, BC V6T 1Z2,
Canada
Joint work with:
Christoph Schwab (Seminar of Applied Mathematics, ETH Zürich, 8092 Zürich,
Switzerland, email: schwab@math.ethz.ch),
Thomas Wihler (Mathematisches Institut, Universität Bern, 3013 Bern, Switzerland,
email: wihler@math.unibe.ch)
We design and analyze hp-version interior penalty (IP) discontinuous Galerkin (dG)
finite element methods for the numerical approximation of linear second order elliptic
boundary-value problems in three dimensional polyhedral domains. The methods are
based on using hexahedral meshes that are geometrically and anisotropically refined
towards corners and edges. Similarly, the local polynomial degrees are increased linearly
and possibly anisotropically away from singularities. We prove that the methods are
well-defined for problems with singular solutions and are stable under the proposed
hp-refinements. Finally, we show that the hp-dG methods lead to exponential rates of
convergence in the number of degrees of freedom for problems with piecewise analytic
data.
References
[1] D. Schötzau, C. Schwab, and T. Wihler: hp-dGFEM for second order elliptic problems
in polyhedra I: Stability and quasioptimality on geometric meshes, submitted.
[2] D. Schötzau, C. Schwab, and T. Wihler: hp-dGFEM for second order elliptic
problems in polyhedra II: Exponential convergence, submitted.
34

Thursday, 14:30–15:25
A Posteriori Error Analysis for Linear Parabolic
PDE based on hp-discontinuous Galerkin
Time-Stepping
Thomas P. Wihler
wihler@math.unibe.ch
Mathematics Institute, University of Bern, Switzerland
Joint work with:
Omar Lakkis (U of Sussex, UK),
Manolis Georgoulis (U of Leicester, UK)
Dominik Schötzau (UBC), Canada
We consider full discretizations in time and space of linear parabolic PDE. More precisely,
in order to discretize the time variable the hp-discontinuous Galerkin time stepping
scheme from [3] is employed. Furthermore, conforming variational methods are
applied to approximate the problem in space.
The focus of the talk is the development of an L 2 (H 1 )-based a posteriori error analysis
(in time and space) for the given class of schemes. Here, the key point is the combination
of an hp-time reconstruction (see [4], cf. also [2]) and of an elliptic reconstruction
in space (see [1]). The resulting a posteriori error estimate is expressed in terms of the
fully discrete solution.
References
[1] C. Makridakis and R. H. Nochetto, Elliptic reconstruction and a posteriori error
estimates for parabolic problems, SIAM Journal on Numerical Analysis, 41 (2003),
No. 4, 1585–1594.
[2] C. Makridakis and R.H. Nochetto, A posteriori error analysis for higher order
dissipative methods for parabolic problems, Numerische Mathematik, 104 (2006),
489–514.
[3] D. Schötzau and C. Schwab, Time discretization of parabolic problems by the hpversion
of the discontinuous Galerkin finite element method, SIAM Journal on
Numerical Analysis, 38 (2000), 837–875.
[4] D. Schötzau and T. P. Wihler, A posteriori error estimation for hp-version timestepping
methods for parabolic partial differential equations, Numerische Mathematik,
115 (2010), no. 3, 475–509.
35

Thursday, 15:30–16:25
Certified Reduced Basis Methods for Integral
Equations with Applications to Electromagnetics
Jan S Hesthaven
Jan.Hesthaven@Brown.edu
Division of Applied Mathematics, Brown University, Providence, RI 02912, USA
Joint work with:
Yvon Maday (Laboratory of Jacques-Louis Lions, University of Paris VI,
Paris, France),
Benjamin Stamm (Department of Mathematics, UC Berkeley, CA 94720, USA),
Shun Zhang (Division of Applied Mathematics, Brown University,
Providence, RI 02912, USA)
The development and application of models of reduced computational complexity is
used extensively throughout science and engineering to enable the fast/real-time/subscale
modeling of complex systems for control, design, prediction purposes of multi-scale
problems, uncertainty quantification etc. Such models, while often successful and of
undisputed value, are, however, often heuristic in nature and the validity and accuracy
of the output is often unknown. This limits the predictive value of these models.
In this talk we shall focus on parametrized models based on integral equations. These
are used widely to model a variety of important problems in acoustics, heat conduction,
and electromagnetics and the analysis and computational techniques have been well
developed during the last decades. However, for many-query situations the direct
solution of these models may be prohibitive and the development of reduced models
of significant value. In the context of integral equations for electromagnetic scattering,
variations over frequency, sources, observations etc, can be substantial, yet a rapid
evaluation has important application in design and detection applications.
We discuss the development of certified reduced basis methods for integral equations to
enable a rapid and error-controlled evaluation of the scattering over parametric variation.
During this talk we outline the basic elements of a certified reduced basis method
and highlight specific issues related to integral equations caused by their non-affine
nature and the need to attend to computational complexity due to large parameter
variations.
We continue by discussing how these reduced models can be combined iteratively to
solve larger multi-object problems without directly assembling the full physical problem.
Time permitting we extend this discussion to the high-dimensional parametrized
problem and discuss ways of achieving accurate parametric compression in the reduced
model using ANOVA expansions.
36

Thursday, 17:00–17:55
Convergence of hp-BEM for the electric field
integral equation
Ralf Hiptmair
hiptmair@sam.math.ethz.ch
ETH Züich
Joint work with:
A. Bespalov (University of Manchester),
N. Heuer (Pontificia Universidad Católica de Chile, Santiago)
We consider the variational formulation of the electric field integral equation (EFIE)
on bounded polyhedral open or closed surfaces. We employ a conforming Galerkin
discretization based on div Γ -conforming Raviart-Thomas boundary elements (BEM)
of locally variable polynomial degree on shape-regular surface meshes. We establish
asymptotic quasi-optimality of Galerkin solutions on sufficiently fine meshes or for
sufficiently high polynomial degree.
References
[1] A. Bespalov, N. Heuer, and R. Hiptmair, Convergence of natural hp-BEM for the
electric field integral equation on polyhedral surfaces,, SIAM J. Numer. Anal., 48
(2010), pp. 1518–1529.
37

Friday, 9:00–9:55
On local refinement in isogeometric analysis
Annalisa Buffa
annalisa@imati.cnr.it
Istituto di Matematica Applicata e Tecnologie Informatiche ’E. Magenes’, CNR,
Via Ferrata 1, Pavia, Italy
Isogeometric methodologies are designed with the aim of improving the connection
between numerical simulation of physical phenomena and the Computer Aided Design
systems. Indeed, the ultimate goal is to eliminate or drastically reduce the approximation
of the computational domain and the re-meshing by the use of the “exact”
geometry directly on the coarsest level of discretization. This is achieved by using
B-Splines or Non Uniform Rational B-Splines for the geometry description as well as
for the representation of the unknown fields. The use of Spline or NURBS functions,
together with isoparametric concepts, results in an extremely successfully idea and
paves the way to many new numerical schemes enjoying features that would be extremely
hard to achieve within a standard finite element framework. One of the big
challenge in the context of isogeometric analysis is the possibility to break the tensor
product structure of Splines and so design adaptive methods. T-splines, introduced
by T. Sederberg et al. in 2004, have been used to this aim in geometric modeling and
design, but their use in isogeometric analysis requires a lot of care.
In this talk, after a short introduction about Isogeometric analysis, I will report on our
experience as regards the use of T-splines as a tool for local refinement.
References
[1] L. Beirao da Veiga, A. Buffa, D. Cho, G. Sangalli, Analysis-Suitable T-splines are
Dual-Compatible. Tech. Rep. IMATI CNR, 2012.
[2] A. Buffa, D. Cho, M. Kumar, Characterization of T-splines with reduced continuity
order on T-meshes, Comput. Methods Appl. Mech. Engrg., Volumes 201204, 1
January 2012, pp. 112-126.
[3] L Beirao da Veiga, A. Buffa, D. Cho, G. Sangalli, Isogeometric analysis using
T-splines on two-patch geometries, Computer Methods in Applied Mechanics and
Engineering, Vol. 200 (21-22), pp.1787-1803, (2011)
38

Friday, 10:00–10:30
Domain decomposition methods
in Isogeometric analysis
Lourenco Beirão da Veiga
lourenco.beirao@unimi.it
Department of Mathematics, University of Milan,
Via Saldini 50, 20133, Milan, Italy.
Joint work with:
Durkbin Cho (University of Milan),
Luca Pavarino (University of Milan),
Simone Scacchi (University of Milan)
NURBS-based isogeometric analysis (IGA in short) [3] is a recent and innovative numerical
methodology for the analysis of PDE problems, that adopts NURBS basis
functions for the description of the discrete space and allows for an exact description
of CAD-type geometries.
The discrete systems produced by isogeometric methods are better conditioned than
the systems produced by standard finite elements or finite differences, but their conditioning
can still degenerates rapidly for decreasing mesh size h, increasing polynomial
degrees p and increasing coefficient jumps in the elliptic operator. Therefore the design
and analysis of efficient iterative solvers for isogeometric analysis is a challenging
research topic, particularly for three-dimensional problems with discontinuous coefficients.
The global high regularity of the adopted NURBS functions introduces a new
set of difficulties to be dealt with, both at the practical and the theoretical level.
In the present talk we will focus on Domain Decomposition [4] methods for Isogeometric
Analysis concentrating our efforts on a model elliptic problem. We will consider both
an Overlapping Schwarz [1] method and a BDDC (i.e. non overlapping) method [2]. In
both cases we will show theoretical condition number bounds in terms of the mesh sizes
h, H (such as the scalability property) and numerical tests that confirm and extend in
terms of the polynomial order p and regularity index k the theoretical predictions.
The theoretical and numerical results of the present talk are strongly encouraging and
show that domain decomposition seems to be a very good choice for preconditioning
and solving problems in Isogeometric Analysis. Clearly more work needs to be accomplished,
both in terms of extension to more complex problems and of development (and
comparison with) different preconditioning and solving schemes.
39

References
[1] L. Beirao da Veiga, D. Cho, L. F. Pavarino, S. Scacchi, Overlapping Schwarz
Methods for Isogeometric Analysis, to appear on Siam J. Numer. Anal. and Preprint
IMATI-CNR (2011).
[2] L. Beirao da Veiga, D. Cho, L. F. Pavarino, S. Scacchi, BDDC preconditioners for
Isogeometric Analysis, submitted for publication and Preprint IMATI-CNR (2012).
[3] J. A. Cottrell, T. J. R. Hughes, Y. Basilevs, Isogeometric Analysis: Toward Integration
of CAD and FEA, Wiley, Chichester, U.K., 2009
[4] A. Toselli, O. B. Widlund, Domain Decomposition Methods: Algorithms and Theory,
Computational Mathematics, Vol. 34. Springer-Verlag, Berlin, 2004.
40

Friday, 10:30–11:00
Isogeometric Collocation Methods for Elasticity
Alessandro Reali
alessandro.reali@unipv.it
Structural Mechanics Department, University of Pavia
Joint work with:
Ferdinando Auricchio (University of Pavia),
Lourenco Beirão da Veiga (University of Milan),
Thomas J.R. Hughes (University of Texas at Austin),
Giancarlo Sangalli(University of Pavia)
Isogeometric Analysis (IGA) is a recent idea, firstly introduced by Hughes et al. [1, 2] to
bridge the gap between Computational Mechanics and Computer Aided Design (CAD).
The key feature of IGA is to extend the finite element method representing geometry by
functions, such as Non-Uniform Rational B-Splines (NURBS), which are used by CAD
systems, and then invoking the isoparametric concept to define field variables. Thus,
the computational domain exactly reproduces the NURBS description of the physical
domain. Moreover, numerical testing in different situations has shown that IGA holds
great promises, with a substantial increase in the accuracy-to-computational-effort ratio
with respect to standard finite elements, also thanks to the high regularity properties
of the employed functions.
In the framework of NURBS-based IGA, collocation methods have been recently proposed
by Auricchio et al. [3], constituting an interesting high-order low-cost alternative
to standard Galerkin approaches. Such techniques have also been successfully applied
to elastostatics and explicit elasodynamics [4].
In this work, after an introduction to isogeometric collocation methods, we move to the
solution of elasticity problems and present in detail the results discussed in [4]. Particular
attention is devoted to the imposition of boundary conditions (of both Dirichlet
and Neumann type) and to the treatment of the multi-patch case. Also the development
of explicit high-order (in space) collocation methods for elastodynamics is here
considered and studied. Several numerical experiments are presented in order to show
the good behavior of these approximation techniques.
References
[1] Hughes, T.J.R., Cottrell, J.A., Bazilevs, Y., Isogeometric analysis: CAD, finite
elements, NURBS, exact geometry, and mesh refinement, Computer Methods in
Applied Mechanics and Engineering, 194, 4135-4195 (2005).
41

[2] Cottrell, J.A., Hughes, T.J.R., Bazilevs, Y., Isogeometric Analysis. Towards integration
of CAD and FEA. Wiley, (2009).
[3] Auricchio, F., Beiro da Veiga, L., Hughes, T.J.R., Reali, A., Sangalli, G., Isogeometric
Collocation Methods, Mathematical Models and Methods in Applied Sciences,
20, 2075-2107 (2010).
[4] Auricchio, F., Beiro da Veiga, L., Hughes, T.J.R., Reali, A., Sangalli, G., Isogeometric
collocation for elastostatics and explicit dynamics, ICES Report (2012).
42

Friday, 11:30–12:25
Isogeometric discretizations of the Stokes problem
Giancarlo Sangalli
giancarlo.sangalli@unipv.it
Dipartimento di Matematica, Università di Pavia
and IMATI-CNR “Enrico Magenes”, Pavia
Joint work with:
Andrea Bressan (Dipartimento di Matematica, Università di Pavia),
Annalisa Buffa (IMATI-CNR “Enrico Magenes”, Pavia),
Carlo de Falco (MOX, Milano),
Rafael Vázquez (IMATI-CNR “Enrico Magenes”, Pavia)
Isogeometric analysis has been introduced in 2005 by T.J.R. Hughes and co-authors
as a novel technique for the discretization of Partial Differential Equations (PDE).
This technique is having a growing impact on several fields, from fluid dynamics, to
structural mechanics, and electromagnetics. A comprehensive reference is the book
[6]. Isogeometric methodologies adopt B-Splines or Non-Uniform Rational B-Splines
(NURBS) functions for the geometry description as well as for the representation of
the unknown fields. Splines and NURBS offer a flexible set of basis functions for
which mesh refinement and degree elevation are very efficient. Beside the fact that
in isogeometric analysis one can directly treat geometries described by Splines and
NURBS parametrizations, these functions are interesting in themselves since they easily
allow global smoothness beyond the classical C 0 -continuity of FEM.
This talk presents the isogeometric methods for the Stokes studied in [2, 3].
In [2] we have analyzed discretizations that extend the Taylor-Hood element, wellknown
in the finite element context, to the isogeometric context. These isogeometric
Taylor-Hood elements are based on p + 1 degree NURBS velocity approximation and p
degree NURBS pressure approximation. The novelty, in the isogeometric framework, is
that C r global regularity is allowed, up to r = p−1. The stability analysis is performed.
In [3] we have proposed a smooth Raviart-Thomas isogeometric discretization that
provides an exact divergence-free discrete solution. This is possible thanks to the
commuting de Rham digram property that this element fulfils (see [5, 4]).
References
[1] Andrea Bressan. Isogeometric regular discretization for the Stokes problem. IMA
Journal of Numerical Analysis, 2010.
43

[2] Andrea Bressan, Giancarlo Sangalli Isogeometric discretizations of the Stokes problem:
stability analysis by the macroelement technique. IMA Journal of Numerical
Analysis, submitted.
[3] A. Buffa, C. de Falco, and G. Sangalli. Isogeometric analysis: stable elements for
the 2d stokes equation. International Journal for Numerical Methods in Fluids,
65(11-12):1407–1422, 2011.
[4] A. Buffa, J. Rivas, G. Sangalli, and R. Vázquez. Isogeometric discrete differential
forms in three dimensions. SIAM Journal on Numerical Analysis, 49:818, 2011.
[5] A. Buffa, G. Sangalli, and R. Vázquez. Isogeometric analysis in electromagnetics:
B-splines approximation. Computer Methods in Applied Mechanics and Engineering,
199(17-20):1143–1152, 2010.
[6] J.A. Cottrell, T.J.R. Hughes, and Y. Bazilevs. Isogeometric analysis: toward integration
of CAD and FEA. John Wiley & Sons Inc, 2009.
44

Friday, 14:30–15:00
Numerical homogenization procedures in solid
mechanics using the finite cell method
Alexander Düster
alexander.duester@tu-harburg.de
Numerische Strukturanalyse mit Anwendungen in der Schiffstechnik (M-10)
TU Hamburg-Harburg
Joint work with:
H.-G. Sehlhorst, M. Joulaian (TU Hamburg-Harburg), E. Rank (TU München)
A numerical procedure for the homogenization of heterogeneous and foamed materials
is presented. Based on this approach, effective material properties are determined
which can be utilized in finite element computations of engineering structures. The
numerical homogenization strategy is applied to model and compute three-dimensional
composites and sandwich structures which are composed of a foamed core covered by
two faceplates. The homogenization is performed by applying the Finite Cell Method
(FCM) which can be interpreted as a combination of a fictitious domain method with
high-order finite elements. Applying the FCM dramatically simplifies the meshing process,
since a simple Cartesian grid is used and the geometry of the microstructure and
the different material properties are taken care of during the integration of the stiffness
matrices of the cells. In this way, the effort is shifted from mesh generation towards the
numerical integration, which can be performed adaptively in a fully automatic way. It
will be demonstrated by several numerical examples that this approach can be applied
to analyze a computer-aided design of a new heterogeneous material or data obtained
by a three-dimensional computer-tomography of the material of interest.
References
[1] A. Düster, H.-G. Sehlhorst, and E. Rank, Numerical homogenization of heterogeneous
and cellular materials utilizing the Finite Cell Method, Computational
Mechanics, (2012), DOI: 10.1007/s00466-012-0681-2.
[2] A. Düster, J. Parvizian, Z. Yang, and E. Rank, The Finite Cell Method for 3D
problems of solid mechanics, Computer Methods in Applied Mechanics and Engineering,
(2008), 197:3768–3782.
[3] J. Parvizian, A. Düster, and E. Rank, Finite Cell Method: h- and p-extension for
embedded domain problems in Solid Mechanics, Computational Mechanics, (2007),
41:121-133.
45

Friday, 15:00–15:30
Simulating the mechanical response of arteries
by p-FEMs
Zohar Yosibash
zohary@bgu.ac.il
Head-Computational Mechanics Lab, Dept. of Mechanical Engineering,
Ben-Gurion University of the Negev, Beer-Sheva, Israel
Joint work with:
Elad Priel (Dept. of Mechanical Engineering, Ben-Gurion University of the Negev,
Beer-Sheva, Israel)
The healthy human artery wall is a complex biological structure whose mechanical response
is of major interest and attracted a significant amount of research, mainly related
to its passive response. Herein we present a high-order FE method for the treatment
of the highly non-linear, almost-incopressible hyper-elastic constitutive model for the
coupled passive-active response based on a transversely isotropic strain-energy-densityfunction
(SEDF).
More specifically, we first address SEDF of the passive response [1]. The new p-FE algorithms
developed to expedite the numerical computations are presented. Numerical
examples are provided to demonstrate the efficiency of the p-FEMs compared to classical
h-FEMs, and thereafter the influence of the slight compressibility on the results.
Although the active response (when smooth muscle cells are activated) has a significant
influence on the overall mechanical response of the artery wall, it has been scarcely
investigated because of the difficulty of representing such an influence in a constitutive
model and lack of experimental evidence that can validate such models. This active
response is strongly coupled with the passive response because of the mutual interaction
(smooth muscle cells react as a function of the mechanical response). We incorporated
the coupled passive-active response in our p-FE simulations that will also presented
[2].
References
[1] Zohar Yosibash and Elad Priel, p-FEMs for hyperelastic anisotropic nearly incompressible
materials under finite deformations with applications to arteries simulation,
International Journal for Numerical Methods in Engineering, 88, (2011),
1152–1174.
[2] Zohar Yosibash and Elad Priel, Artery active mechanical response: High order
finite element implementation and investigation, Submitted for publication, (2012)
46

Friday, 16:00–16:55
A generalized empirical interpolation method
based on moments and application
Yvon Maday
maday@ann.jussieu.fr
UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris,
France
Joint work with:
Olga Mula (UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions,
F-75005, Paris and CEA Saclay, DEN/DANS/DM2S/SERMA/LENR, France)
The empirical interpolation method introduced in [1] (see also [2]) allows to provide a
simple and efficient construction of an interpolation process on manifold of small Kolmogorov
width. This approach has attracted some attention in the frame of the reduced
basis method and more generally reduced order approximation for the approximation
of nonlinear PDE’s. The application range is nevertheless much larger.
We propose in this talk a generalization of this concept with applications to combined
assimilation-simulation technics for complex phenomenon.
References
[1] Barrault, Maxime and Maday, Yvon and Nguyen, Ngoc Cuong and Patera, Anthony
T. An ‘empirical interpolation’ method: application to efficient reduced-basis
discretization of partial differential equations, Comptes Rendus Mathématique.
Académie des Sciences. Paris, 339 (2004) 9, pp 667–672
[2] Maday, Yvon and Nguyen, Ngoc Cuong and Patera, Anthony T. and Pau, George
S. H. A general multipurpose interpolation procedure: the magic points, Commun.
Pure Appl. Anal., 8, (2009), 1, pp 383–404.
47

List of Participants
Ainsworth Mark University of Strathclyde
Andreev Roman ETH Zürich
Banz Lothar Leibniz Universität Hannover
Bartha Ferenc University of Bergen
Beirão da Veiga Lourenco Università di Milano
Bespalov Alexey University of Manchester
Beuchler Sven Universität Bonn
Bierig Claudio Universität Bonn
Brandsmeier Holger ETH Zürich
Buffa Annalisa CNR, Pavia
Canuto Claudio Politecnico di Torino
Chernov Alexey Universität Bonn
Costabel Martin Université de Rennes 1
Dauge Monique Université de Rennes 1
Demkowicz Leszek University of Texas at Austin
Düster Alexander TU Hamburg-Harburg
Egger Herbert TU München
Gittelson Claude Jeffrey Purdue University
Hackbusch Wolfgang MPI Leipzig
Hakula Harri Aalto University School of Science and Technology
Hesthaven Jan S. Brown University
Heuveline Vincent KIT, Karlsruhe
Hiptmair Ralf ETH Zürich
Houston Paul University of Nottingham
Klose Roland CST AG, Darmstadt
Litvinenko Alexander TU Braunschweig
Maday Yvon Université Pierre et Marie Curie
Maischak Matthias Brunel University, Uxbridge
Melenk Jens Markus TU Wien
Merabet Ismail Université de Valenciennes
Nicaise Serge Université de Valenciennes
Nobile Fabio EPFL Lausanne
Pham Duong Thanh Universität Bonn
Quarteroni Alfio EPFL Lausanne
Rank Ernst TU München
Reali Alessandro Università di Pavia
Reinarz Anne Universität Bonn
48

Sangalli Giancarlo Università di Pavia
Schmidt Kersten TU Berlin
Schöberl Joachim TU Wien
Schötzau Dominik University of British Columbia, Vancouver
Schröder Andreas HU Berlin
Schwab Christoph ETH Zürich
Sherwin Spencer J. Imperial College London
Sloan Ian H. UNSW Sydney
Sprungk Bjoern TU Bergakademie Freiberg
Stamm Benjamin University of California, Berkeley
Stephan Ernst P. Leibniz Universität Hannover
Valli Alberto Università di Trento
Weggler Lucy Universität des Saarlandes
Wihler Thomas Universität Bern
Xiu Dongbin Purdue University
Yosibash Zohar Ben-Gurion University, Beer-Sheva
Zaglmayr Sabine CST AG, München
49