Here - Hausdorff Center for Mathematics - Universität Bonn
Here - Hausdorff Center for Mathematics - Universität Bonn
Here - Hausdorff Center for Mathematics - Universität Bonn
Create successful ePaper yourself
Turn your PDF publications into a flip-book with our unique Google optimized e-Paper software.
Workshop<br />
High-Order Numerical Approximation<br />
<strong>for</strong> Partial Differential Equations<br />
<strong>Hausdorff</strong> <strong>Center</strong> <strong>for</strong> <strong>Mathematics</strong><br />
University of <strong>Bonn</strong><br />
February 6–10, 2012<br />
Organizers<br />
Alexey Chernov (University of <strong>Bonn</strong>)<br />
Christoph Schwab (ETH Zurich)
Program overview<br />
Time Monday Tuesday Wednesday Thursday Friday<br />
8:00 Registration<br />
8:40 Opening<br />
9:00<br />
Stephan Sloan Ainsworth Demkowicz Buffa<br />
10:00<br />
10:30<br />
Canuto Nobile Sherwin Houston<br />
Beirão<br />
da Veiga<br />
Reali<br />
11:00<br />
Coffee break<br />
11:30<br />
12:00<br />
Quarteroni Gittelson Schöberl Schötzau Sangalli<br />
Bespalov<br />
12:30<br />
Lunch break<br />
14:30<br />
15:00<br />
15:30<br />
16:00<br />
Dauge Xiu Wihler Düster<br />
Excursions:<br />
Arithmeum<br />
Melenk Litvinenko Hesthaven<br />
or<br />
Maischak<br />
16:30<br />
Coffee break<br />
17:00<br />
–18:00<br />
Botanic<br />
Gardens<br />
Coffee break<br />
Hackbusch Rank Hiptmair<br />
Free Time<br />
Yosibash<br />
Coffee break<br />
Maday<br />
20:00 Dinner<br />
3
High-Order Numerical Approximation<br />
<strong>for</strong> Partial Differential Equations<br />
Over the last decades high-order numerical approximation <strong>for</strong> PDEs have become wellestablished<br />
tools <strong>for</strong> the efficient, highly accurate numerical solution of (boundary)<br />
integral equations and partial differential equations in computational science and engineering;<br />
prominent examples are hp- and spectral element methods in application areas<br />
such as computational fluid dynamics, computational climate modelling, to name but<br />
a few. Their theoretical convergence analysis and adaptive hp- and spectral element<br />
versions currently experience strong development. New application areas <strong>for</strong> spectral<br />
methods are infinite dimensional Fokker-Planck equations, stochastic partial differential<br />
equations, uncertainty quantification and isogeometric analysis. <strong>Here</strong>, analysis is<br />
missing to a large extent and is subject of current research.<br />
The aim of the Workshop is to bring together leading experts in analysis and implementation<br />
of High-Order Methods <strong>for</strong> PDEs, thereby stimulating an intensive idea<br />
exchange and fruitful interactions.<br />
Workshop Venue<br />
All talks of the workshop will take place at the <strong>Mathematics</strong> <strong>Center</strong>, Endenicher Allee 60,<br />
in the Lipschitz Hall (Room 1.016), first floor (European counting). Please feel free to<br />
use Rooms 1.007 and 1.008 as well as the blackboard in the Plücker Hall (Room 1.015)<br />
<strong>for</strong> discussions. Please feel free to use the mathematics library in the ground floor. It<br />
has a large collection of books and journals.<br />
Lipschitz Hall<br />
1.016<br />
(lecture room)<br />
Plücker Hall<br />
1.015<br />
(coffee<br />
breaks)<br />
Stairs up<br />
Elevator<br />
Stairs down<br />
(Library /<br />
Ground floor)<br />
Discussion<br />
Room<br />
1.007<br />
Main<br />
Entrance<br />
(1st Floor)<br />
Discussion<br />
Room<br />
1.008<br />
Endenicher Alle<br />
4
Getting there<br />
The most convenient way to get to the workshop venue from the hotel “My Poppelsdorf”<br />
is walking (about 10-15 min). Alternatively, you might take a bus 631 from<br />
the stop “Poppelsdorfer Platz” to the stop “Kaufmannstrasse” (4 stops, goes every 30<br />
min.) or a taxi.<br />
P<br />
P<br />
AS <strong>Bonn</strong>-<br />
Endenich<br />
U<br />
ENDENICH<br />
Röckumstr.<br />
Flodelingsweg<br />
Kapellenstr.<br />
Immenburgstr.<br />
WESTSTADT<br />
Endenicher Str.<br />
AS <strong>Bonn</strong>-<br />
Poppelsdorf<br />
A 565<br />
Thomastr.<br />
Endenicher Allee 60<br />
Workshop Venue<br />
Meckenheimer Allee 171<br />
Botanic Gardens<br />
Wallfahrtsweg 4<br />
Hotel „My Poppelsdorf”<br />
Sebastianstr.<br />
NORDSTADT<br />
Heerstr.<br />
Wittelsbacherring<br />
Haydnstr.<br />
Bornheimer St r.<br />
Kaufmannstr. Nußallee<br />
Kreuzberg-<br />
Humboldtstr.<br />
Richard-Wagner-<br />
Str.<br />
Endenicher Allee<br />
Carl-Troll-Str.<br />
Wegelerstr.<br />
Katzenburgweg<br />
Alter<br />
Friedhof<br />
Sternenburgstr.<br />
Clemens-August-Str.<br />
Rabinstr.<br />
Herwarthstr.<br />
Colmantstr.<br />
Beethovenstr.<br />
Stadthaus<br />
Poppelsdorfer<br />
Schloss<br />
Botanische<br />
Gärten<br />
H H<br />
H<br />
Baumschulallee<br />
Prinz-<br />
Beringstr.<br />
weg<br />
Kirschallee<br />
Maxstr.<br />
Th.-Mann-Str.<br />
Meckenheimer Allee<br />
H Hbf<br />
DB<br />
U<br />
POPPELSDORF<br />
Post<br />
Quantiusstr.<br />
Sternstr.<br />
Gang<br />
Poststr.<br />
olfstr.<br />
Poppelsdorfer Allee<br />
str.<br />
Maximilianstr.<br />
H<br />
H<br />
H<br />
SÜDSTADT<br />
Reuterstr .<br />
Maarflach<br />
Kaiserpl.<br />
<strong>Bonn</strong>er Talweg<br />
<strong>Bonn</strong>gasse<br />
Remigiusstr.<br />
Sandkaule<br />
Ox<strong>for</strong>d-<br />
<strong>Bonn</strong>gasse 30<br />
Restaurant „Im Stiefel”<br />
str. B.-v.-Suttner-<br />
H Platz<br />
str.<br />
Jagdweg Kurfürsten-<br />
Venusbergstr.<br />
Bismarckstr.<br />
Königweg<br />
Argelander-<br />
Goeben-<br />
Wenzelg.<br />
Markt<br />
Am Hof<br />
Uni-<br />
Hauptgebäude<br />
Rathaus<br />
U<br />
Hofgarten<br />
Kaiserstr.<br />
Albert-<br />
str.<br />
str.<br />
Wallfahrtsweg<br />
Wesselstr.<br />
A. d.<br />
Schloßkirche<br />
Str.<br />
Belderberg<br />
Regina-Pacis-<br />
Münsterpl.<br />
Friedrich-<br />
Friedensplatz<br />
Franziskanerstr.<br />
Am<br />
Stockenstr.<br />
H<br />
Weg<br />
Am<br />
Boeselagerhof<br />
Giergasse<br />
Opernhaus<br />
Hofgarten<br />
W eber-<br />
Nassestr.<br />
Konviktstr.<br />
Schumannstr.<br />
Adenauerallee<br />
Lennéstr. 2<br />
Arithmeum<br />
Lennéstr.<br />
Brassertufer<br />
Koblenzer Tor<br />
Kennedybrücke<br />
H<br />
An der<br />
RHEIN<br />
Erste<br />
Fährg.<br />
Rathenauufer<br />
U<br />
str.<br />
Lessingstr. Arndtstr.<br />
Elisabeth kirche<br />
0 500 m<br />
© Geographisches Institut der Universität <strong>Bonn</strong><br />
Lunch and coffee breaks<br />
Coffee will be served during the coffee breaks in the Plücker Hall (Room 1.015), next to<br />
the Lipschitz Hall where all talks will take place. For the lunch you might go to a place<br />
of your choice. Many restaurants are located in the Clemens-August street (you should<br />
have passed it on your way from the hotel “My Poppelsdorf” to the Workshop Venue)<br />
or in the city center. Another option is the student restaurant (exit the <strong>Mathematics</strong><br />
<strong>Center</strong>, cross the street and turn to the right).<br />
5
Internet access via WLAN<br />
We offer a free Internet access via WLAN in the Workshop Venue. For this, you need<br />
an access certificate. You should have received it via email in the beginning of January.<br />
If you have difficulties with the Internet connection, please contact the registration desk<br />
or the organizers.<br />
Excursion on Wednesday<br />
The Wednesday’s afternoon is free and we welcome you to join one of two excursions:<br />
• Arithmeum (http://www.arithmeum.uni-bonn.de), a museum of historical<br />
mechanical calculating machines, historical arithmetic books, etc.<br />
• Green Houses of the Botanic Gardens of the University of <strong>Bonn</strong><br />
Both excursions start at 15:00 and will be accompanied by an English guide. You will<br />
be expected at 15:00 at the entrance to Arithmeum (Lenné Str. 2) or Botanic Gardens<br />
(Meckenheimer Allee 171). The locations are designated on the city plan.<br />
The excursion in Arithmeum is free. A small fee (several Euros, depending on the<br />
number of participants) applies <strong>for</strong> the excursion in the Botanic Gardens. We kindly<br />
ask you to in<strong>for</strong>m us during the registration, or by Monday evening the latest, which<br />
excursion would you like to join.<br />
Dinner on Thursday<br />
On Thursday’s evening we welcome you to join us about 20:00 <strong>for</strong> the Workshop Dinner<br />
at the “typical” restaurant Im Stiefel in the center of <strong>Bonn</strong> (<strong>Bonn</strong>gasse 30). Un<strong>for</strong>tunately,<br />
due to our budget rules we are not able to cover the costs <strong>for</strong> the Dinner.<br />
Everybody will have to pay himself / herself. The food prices (excluding drinks) range<br />
between 8 and 14 Euro.<br />
We kindly ask you to in<strong>for</strong>m us during the registration, or by Monday evening the<br />
latest, if you wish to join us <strong>for</strong> the Workshop Dinner.<br />
Acknowledgement<br />
The organizers would like to thank the <strong>Hausdorff</strong> <strong>Center</strong> <strong>for</strong> <strong>Mathematics</strong> <strong>for</strong> the<br />
generous financial support, which made this Workshop possible.<br />
Special thanks go to Laura Siklossy and other members of the HIM and HCM Administration<br />
& Support teams <strong>for</strong> the strong an continuous support in all administrative<br />
issues. Many thanks to Gunder-Lily Sievert and her team <strong>for</strong> organizing the coffee<br />
breaks. We also thank Anne Reinarz, Duong Pham and Claudio Bierig <strong>for</strong> helping us<br />
with some practical issues be<strong>for</strong>e and during the Workshop.<br />
6
Monday, February 6<br />
Detailed program<br />
9:00–9:55 Ernst P. Stephan<br />
hp-adaptive DG-FEM <strong>for</strong> Parabolic Obstacle Problems<br />
10:00–10:55 Claudio Canuto<br />
Adaptive Fourier-Galerkin Methods<br />
11:00–11:30 Coffee break<br />
11:30–12:25 Alfio Quarteroni<br />
Discontinuous approximation of elastodynamics equations<br />
12:30–14:30 Lunch break<br />
14:30–15:25 Monique Dauge<br />
Weighted analytic regularity in polyhedra<br />
15:30–16:25 Jens Markus Melenk<br />
Stability and convergence of Galerkin discretizations of the Helmholtz equation<br />
16:30–17:00 Coffee break<br />
17:00–17:55 Wolfgang Hackbusch<br />
Selfadaptive Compression in Tensor Calculus<br />
Tuesday, February 7<br />
9:00–9:55 Ian H. Sloan<br />
PDE with random coefficients as a problem in high dimensional integration<br />
10:00–10:55 Fabio Nobile<br />
Stochastic Polynomial approximation of PDEs with random coefficients<br />
11:00–11:30 Coffee break<br />
11:30–12:00 Claude J. Gittelson<br />
Potential and Limitations of Sparse Tensor Product Stochastic Galerkin Methods<br />
12:00–12:30 Alex Bespalov<br />
A priori error analysis of stochastic Galerkin mixed finite element methods<br />
12:30–14:30 Lunch break<br />
14:30–15:25 Dongbin Xiu<br />
A Flexible Stochastic Collocation Algorithm on Arbitrary Nodes via Interpolation<br />
15:30–16:00 Alexander Litvinenko<br />
Computation of maximum norm and level sets in the canonical tensor <strong>for</strong>mat with<br />
application to SPDE<br />
16:00–16:30 Matthias Maischak<br />
p-FEM and p-BEM – achieving high accuracy<br />
16:30–17:00 Coffee break<br />
17:00–17:55 Ernst Rank<br />
To mesh or not to mesh: High order numerical simulation <strong>for</strong> complex structures<br />
7
Wednesday, February 8<br />
9:00–9:55 Mark Ainsworth<br />
Bernstein-Bézier finite elements of arbitrary order and optimal assembly procedures<br />
10:00–10:55 Spencer Sherwin<br />
From h to p efficiently: balancing high or low order approximations with accuracy<br />
11:00–11:30 Coffee break<br />
11:30–12:25 Joachim Schöberl<br />
Domain Decomposition Preconditioning <strong>for</strong> High Order Hybrid Discontinuous Galerkin<br />
Methods on Tetrahedral Meshes<br />
12:30–14:30 Lunch break<br />
14:30–18:00 Excursions to “Arithmeum” or Botanic Gardens, free time<br />
Thursday, February 9<br />
9:00–9:55 Leszek Demkowicz<br />
DPG method with optimal test functions. A progress report.<br />
10:00–10:55 Paul Houston<br />
Application of hp-Adaptive Discontinuous Galerkin Methods to Bifurcation Phenomena<br />
in Pipe Flows<br />
11:00–11:30 Coffee break<br />
11:30–12:25 Dominik Schötzau<br />
Exponential convergence of hp-version discontinuous Galerkin methods <strong>for</strong> elliptic problems<br />
in polyhedral domains<br />
12:30–14:30 Lunch break<br />
14:30–15:25 Thomas P. Wihler<br />
A Posteriori Error Analysis <strong>for</strong> Linear Parabolic PDE based on hp-discontinuous Galerkin<br />
Time-Stepping<br />
15:30–16:25 Jan S. Hesthaven<br />
Certified Reduced Basis Methods <strong>for</strong> Integral Equations with Applications to Electromagnetics<br />
16:30–17:00 Coffee break<br />
17:00–17:55 Ralf Hiptmair<br />
Convergence of hp-BEM <strong>for</strong> the electric field integral equation<br />
8
Friday, February 10<br />
9:00–9:55 Annalisa Buffa<br />
On local refinement in isogeometric analysis<br />
10:00–10:30 Lourenco Beirão da Veiga<br />
Domain decomposition methods in Isogeometric analysis<br />
10:30–11:00 Alessandro Reali<br />
Isogeometric Collocation Methods <strong>for</strong> Elasticity<br />
11:00–11:30 Coffee break<br />
11:30–12:25 Giancarlo Sangalli<br />
Isogeometric discretizations of the Stokes problem<br />
12:30–14:30 Lunch break<br />
14:30–15:00 Alexander Düster<br />
Numerical homogenization procedures in solid mechanics using the finite cell method<br />
15:00–15:30 Zohar Yosibash<br />
Simulating the mechanical response of arteries by p-FEMs<br />
15:30–16:00 Coffee break<br />
16:00–16:55 Yvon Maday<br />
A generalized empirical interpolation method based on moments and application<br />
9
Monday, 9:00–9:55<br />
hp-adaptive DG-FEM <strong>for</strong> Parabolic<br />
Obstacle Problems<br />
Ernst P. Stephan<br />
stephan@ifam.uni-hannover.de<br />
Insitut für Angewandte Mathematik (IfAM), Leibniz Universität Hannover (LUH),<br />
Germany<br />
Joint work with:<br />
Lothar Banz (IfAM, LUH, Germany)<br />
Parabolic obstacle problems find applications in the financial markets <strong>for</strong> pricing American<br />
put options. We present a mixed method and an equivalent variational inequality<br />
method, both based on an hp-interior penalty DG (IPDG) method combined with<br />
an hp-time DG (TDG) method, to solve parabolic obstacle problems approximatively<br />
[1]. The contact conditions are resolved by a biorthogonal Lagrange multiplier and<br />
are component-wise decoupled. These decoupled contact conditions are equivalent to<br />
finding the root of a non-linear complementary function. This non-linear problem can<br />
in turn be solved efficiently by a semi-smooth Newton method. For the hp-adaptivity<br />
a p-hierarchical error estimator in conjunction with a local analyticity estimate is employed,<br />
where the latter decides weather an h or p-refinment should be carried out. For<br />
the stationary problem, this leads to exponential convergence, and <strong>for</strong> the instationary<br />
problem to greatly improved convergence rates.<br />
Numerical experiments are given demonstrating the strengths and limitations of the<br />
approaches.<br />
References<br />
[1] L. Banz hp-Finite Element and Boundary Element Methods <strong>for</strong> Elliptic, Elliptic<br />
Stochastic, Parabolic and Hyperbolic Obstacle and Contact Probelems, PhD-Thesis<br />
(submitted).<br />
10
Monday, 10:00–10:55<br />
Adaptive Fourier-Galerkin Methods<br />
Claudio Canuto<br />
claudio.canuto@polito.it<br />
Politecnico di Torino<br />
Joint work with:<br />
R.H. Nochetto (University of Maryland),<br />
M. Verani (Politecnico di Milano)<br />
The design of adaptive spectral-element (or h-p) discretization algorithms <strong>for</strong> elliptic<br />
self-adjoint problems relies on the dynamic interplay between two fundamental stages:<br />
the refinement of the geometric decomposition into elements and the enrichment of the<br />
local basis within an element. While the <strong>for</strong>mer stage is by now well understood in<br />
terms of practical realization and optimality properties, less theoretical attention has<br />
been devoted to the latter one.<br />
With the aim of shedding light on this topic, we focus on what happens in a single element<br />
of the decomposition. In order to reduce at a minimum the technical burdens, we<br />
actually assume periodic boundary conditions on the d-dimensional box Ω = (0, 2π) d ,<br />
in order to exploit the orthogonality properties of the Fourier basis. Thus, we consider<br />
a fully adaptive Fourier method, in which at each iteration of the adaptive algorithm<br />
the Galerkin solution is spanned by a dynamically selected, finite subset of the whole<br />
set of Fourier basis functions. The active set is determined by looking at a fixed fraction<br />
of largest (scaled) Fourier coefficients of the residual, according to the bulk-chasing (or<br />
Dörfler marking) philosophy. The algorithm is proven to be convergent; in addition,<br />
exploiting the concentration properties of the inverse of the elliptic operator represented<br />
in the Fourier basis, one can show that the error reduction factor per iteration<br />
tends to 0 as the bulk-chasing fraction tends to 1.<br />
After convergence has been established, one is faced with the issue of optimality. This<br />
leads to the comparison of the adaptive Galerkin solution spanned by, say, N Fourier<br />
modes, with the best N-term approximation of the exact solution. Consequently, we<br />
are led to introduce suitable sparsity classes of periodic H 1 -functions <strong>for</strong> which the best<br />
N-term approximation error fulfils a prescribed decay as N tends to infinity. These<br />
classes can also be characterized in terms of behavior of the rearranged sequence of the<br />
(normalized) Fourier coefficients of the functions.<br />
If the best N-term approximation error of the exact solution decays at an algebraic<br />
rate (this occurs if the solution belongs to a certain “oblique” scale of Besov spaces of<br />
periodic functions, corresponding to a finite regularity), then the arguments developed<br />
by Cohen, Dahmen and DeVore and by Stevenson et al in the framework of wavelet<br />
bases apply to our situation as well, and one can establish the optimality of the approximation<br />
without coarsening. The crucial ingredients in the analysis are the minimality<br />
11
property of the active set of degrees of freedom determined by bulk-chasing, and a<br />
geometric-series argument (essentially, the estimated number of degrees of freedom<br />
added at each iteration is comparable to the total number of degrees of freedom added<br />
in all previous iterations).<br />
On the other hand, the case of a solution having (local) infinite-order or analytical<br />
regularity is quite significant in dealing with spectral (spectral-element) methods. For<br />
the Navier-Stokes equations, regularity results in Gevrey classes have been first established<br />
by Foias and Temam; in these classes, the best N-term approximation error of<br />
a function decays precisely at a sub-exponential or exponential rate. Then, the analysis<br />
of an adaptive strategy becomes much more delicate, as the previous arguments<br />
fail to apply. For instance, even <strong>for</strong> a linear operator with analytic ceofficients, it is<br />
shown that the Gevrey sparsity class of the residual is different from (actually, worse<br />
than) the one of the exact solution. There<strong>for</strong>e, we devise and analyse a new version<br />
of the adptive algorithm considered be<strong>for</strong>e: since the choice of the next set of active<br />
degrees of freedom is made on the basis of the sparsity of the residual, we incorporate<br />
a coarsening step in order to get close to the sparsity pattern of the solution.<br />
The extension of some of the previous results to the case of non-periodic (Legendre)<br />
expansions will also be considered.<br />
12
Monday, 11:30–12:25<br />
Discontinuous approximation<br />
of elastodynamics equations<br />
Alfio Quarteroni<br />
alfio.quarteroni@epfl.ch<br />
MATHICSE, EPFL, Lausanne and MOX, Politecnico di Milano, Milan<br />
Joint work with:<br />
P.Antonietti, I.Mazzieri and F.Rapetti<br />
The possibility of inferring the physical parameter distribution of the Earth’s substratum,<br />
from in<strong>for</strong>mation provided by elastic wave propagations, has increased the interest<br />
towards computational seismology. Recent developments have been focused on spectral<br />
element methods. The reason relies on their flexibility in handling complex geometries,<br />
retaining the spatial exponential convergence <strong>for</strong> locally smooth solutions and a natural<br />
high level of parallelism. In this talk, we consider a Discontinuous Galerkin spectral<br />
element method (DGSEM) as well as discontinuous mortar methods (DMORTAR) to<br />
simulate seismic wave propagations in three dimensional heterogeneous media. The<br />
main advantage with respect to con<strong>for</strong>ming discretizations as those based on Spectral<br />
Element Method is that DG and Mortar discretizations can accommodate discontinuities,<br />
not only in the parameters, but also in the wavefield, while preserving the energy.<br />
The domain of interest Ω is assumed to be union of polygonal substructures Ω i . We<br />
allow this substructure decomposition to be geometrically non-con<strong>for</strong>ming. Inside each<br />
substructure Ω i , a con<strong>for</strong>ming high order finite element space associated to a partition<br />
T hi (Ω i ) is introduced. We allow the use of different polynomial approximation degrees<br />
within different substructures. Applications to simulate benchmark problems as well<br />
as realistic seismic wave propagation processes are presented.<br />
13
Monday, 14:30–15:25<br />
Weighted analytic regularity in polyhedra<br />
Monique Dauge<br />
Monique.Dauge@univ-rennes1.fr<br />
IRMAR, CNRS and Université de Rennes 1, FRANCE<br />
Joint work with:<br />
Martin Costabel (IRMAR, Université de Rennes 1, FRANCE),<br />
Serge Nicaise (LAMAV, Université de Valenciennes, FRANCE)<br />
Elliptic boundary value problems with analytic data (coefficients, domain and right<br />
hand sides) are analytic up to the boundary [5]. Such a regularity allows exponential<br />
convergence of the p-version of the finite element method. This result does not hold in<br />
the same <strong>for</strong>m in domains with edges and corners.<br />
In ca. 1988 Babuška and Guo started a program relying on three ideas<br />
1. The p-version should be replaced by the hp-version of finite elements,<br />
2. The hp-version is exponentially converging if solutions belong to certain weighted<br />
analytic classes (which they call countably normed spaces),<br />
3. The solutions to standard elliptic boundary problems belong to such analytic<br />
classes.<br />
They proved results covering these three points <strong>for</strong> bi-dimensional problems (Laplace,<br />
Lamé). They started the investigation of three-dimensional problems in [2, 3, 4]. But<br />
three-dimensional problems have a level of complexity higher than 2D, <strong>for</strong> two reasons<br />
a) The hp-version requires anisotropic refinement along edges to prevent a blowing<br />
up of the number of degrees of freedom. Exponential convergence with such discretization<br />
will be obtain only if solutions belong to anisotropic analytic weighted<br />
spaces.<br />
b) The edge behavior combine in a non-trivial way at each corner.<br />
In our paper [1], we prove anisotropic weighted regularity <strong>for</strong> a full class of coercive<br />
variational problems, homogeneous with constant coefficients. This completes <strong>for</strong> the<br />
first time point 3. of the Babuška-Guo program in polyhedra. In this talk, I present the<br />
results of this paper. The case of Laplace-Dirichlet and Laplace-Neumann problems will<br />
serve as threads along our way in the various possible combinations of weights in twoand<br />
three-dimensional domains, and the various corresponding statements. General<br />
statements will be provided as well.<br />
14
References<br />
[1] M. Costabel, M. Dauge, and S. Nicaise, Analytic Regularity <strong>for</strong> Linear Elliptic<br />
Systems in Polygons and Polyhedra, Math. Models Methods Appl. Sci., 22 (2012),<br />
No. 8.<br />
arXiv: http://fr.arxiv.org/abs/1112.4263.<br />
[2] B. Guo. The h-p version of the finite element method <strong>for</strong> solving boundary value<br />
problems in polyhedral domains. In M. Costabel, M. Dauge, S. Nicaise, editors,<br />
Boundary value problems and integral equations in nonsmooth domains (Luminy,<br />
1993), pages 101–120. Dekker, New York 1995.<br />
[3] B. Guo, I. Babuška. Regularity of the solutions <strong>for</strong> elliptic problems on nonsmooth<br />
domains in R 3 . I. Countably normed spaces on polyhedral domains. Proc. Roy.<br />
Soc. Edinburgh Sect. A, 127 (1997), No. 1, 77–126.<br />
[4] B. Guo, I. Babuška. Regularity of the solutions <strong>for</strong> elliptic problems on nonsmooth<br />
domains in R 3 . II. Regularity in neighbourhoods of edges. Proc. Roy. Soc. Edinburgh<br />
Sect. A, 127 (1997), No. 3, 517–545.<br />
[5] C. B. Morrey, Jr., L. Nirenberg. On the analyticity of the solutions of linear elliptic<br />
systems of partial differential equations. Comm. Pure Appl. Math. 10 (1957), 271–<br />
290.<br />
15
Monday, 15:30–16:25<br />
Stability and convergence of Galerkin<br />
discretizations of the Helmholtz equation<br />
J.M. Melenk<br />
melenk@tuwien.ac.at<br />
Vienna University of Technology<br />
Joint work with:<br />
Stefan Sauter (University of Zurich),<br />
Maike Löhndorf (Kapsch TrafficCom)<br />
We consider boundary value problems <strong>for</strong> the Helmholtz equation at large wave numbers<br />
k. In order to understand how the wave number k affects the convergence properties<br />
of discretizations of such problems, we develop a regularity theory <strong>for</strong> the Helmholtz<br />
equation that is explicit in k. At the heart of our analysis is the decomposition of solutions<br />
into two components: the first component is an analytic, but highly oscillatory<br />
function and the second one has finite regularity but features wavenumber-independent<br />
bounds.<br />
This new understanding of the solution structure opens the door to the analysis of<br />
discretizations of the Helmholtz equation that are explicit in their dependence on the<br />
wavenumber k. As a first example, we show <strong>for</strong> a con<strong>for</strong>ming high order finite element<br />
method that quasi-optimality is guaranteed if (a) the approximation order p is selected<br />
as p = O(log k) and (b) the mesh size h is such that kh/p is small.<br />
As a second example, we consider combined field boundary integral equation arising<br />
in acoustic scattering. Also <strong>for</strong> this example, the same scale resolution conditions as<br />
in the high order finite element case suffice to ensure quasi-optimality of the Galekrin<br />
discretization.<br />
16
Monday, 17:00–17:55<br />
Selfadaptive Compression in Tensor Calculus<br />
Wolfgang Hackbusch<br />
wh@mis.mpg.de<br />
Max-Planck-Institut Mathematik in den Naturwissenschaften<br />
Inselstr. 22, D-04103 Leipzig<br />
The principle behind high-order methods is the wish to obtain high accuracy by a<br />
small number of degrees of freedom. This leads, e.g., to the hp-method, where still<br />
the question arises how the step sizes and polynomial degrees are to be chosen optimally.<br />
A further question is whether polynomials should be replaced by another class<br />
of functions. A similar situation occurs <strong>for</strong> wavelet approaches.<br />
The tensor approaches <strong>for</strong> functions in product domains (e.g, in R d ) start usually with a<br />
regular grid as known from old-fashioned difference methods. This seems to contradict<br />
the modern approaches. However, the algorithms in tensor calculus are essentially<br />
based on compression principle. In the best case, the total data size n d (n grid points<br />
in each of the d directions) can be compressed to O(d · log n · log(1/ε)) in order to<br />
obtain an approximation of accuracy ε. It turns out that the obtainable compression<br />
is at least as good as obtainable by hp-methods (or other high-order) methods. The<br />
essential difference is that compression methods in tensor calculus find the suitable<br />
distributions of h and p automatically and even chooses the suitable class of functions<br />
in a selfadaptive way.<br />
References<br />
[1] Hackbusch, Wolfgang: Convolution of hp-functions on locally refined grids. IMA J.<br />
Numer. Anal. 29, 960–985 (2009)<br />
[2] Hackbusch, Wolfgang: Tensor spaces and numerical tensor calculus. SCM 42,<br />
Springer Berlin (2012)<br />
[3] Khoromskij, Boris: O(d log N)-quantics approximation of N − d tensors in highdimensional<br />
numerical modeling. Constr. Approx. (2011). To appear<br />
[4] Oseledets, Ivan V.: Approximation of matrices using tensor decomposition. SIAM<br />
J. Matrix Anal. Appl. 31, 2130–2145 (2010)<br />
17
Tuesday, 9:00–9:55<br />
PDE with random coefficients as a problem<br />
in high dimensional integration<br />
Ian H Sloan<br />
i.sloan@unsw.edu.au<br />
University of New South Wales<br />
Joint work with:<br />
Frances, Y Kuo (University of New South Wales),<br />
Christoph Schwab (ETH Zurich)<br />
This talk is concerned with the use of quasi-Monte Carlo methods combined with finiteelement<br />
methods to handle an elliptic PDE with a random field as a coefficient. A number<br />
of groups have considered such problems (under headings such as polynomial chaos,<br />
stochastic Galerkin and stochastic collocation) by re<strong>for</strong>mulating them as deterministic<br />
problems in a high dimensional parameter space, where the dimensionality comes from<br />
the number of random variables needed to characterise the random field. In a recent<br />
paper we have treated a problem of this kind as one of infinite-dimensional integration<br />
- where integration arises because a multi-variable expected value is a multidimensional<br />
integral - together with finite- element approximation in the physical space. We use<br />
recent developments in the theory and practice of quasi-Monte Carlo integration rules<br />
in weighted Hilbert spaces, through which rules with optimal properties can be constructed<br />
once the weights are known. The novel feature of this work is that <strong>for</strong> the first<br />
time we are able to design weights <strong>for</strong> the weighted Hilbert space that achieve what is<br />
believed to be the best possible rate of convergence, under conditions on the random<br />
field that are exactly the same as in a recent paper by Cohen, DeVore and Schwab on<br />
best N-term approximation <strong>for</strong> the same problem.<br />
References<br />
[1] A. Cohen, R. De Vore and C. Schwab Convergence rates of best N -term apporximations<br />
<strong>for</strong> a class of elliptic sPDEs, Foundations of Computational <strong>Mathematics</strong>,<br />
10 (2010), 615–646.<br />
[2] F. Y. Kuo, C. Schwab and I.H. Sloan Quasi-Monte Carlo finite element methods <strong>for</strong><br />
a class of elliptic partial differential equations with random coefficients, submitted,<br />
2011.<br />
18
Tuesday, 10:00–10:55<br />
Stochastic Polynomial approximation of PDEs<br />
CSQI MATHICSE,<br />
with random coefficients<br />
Fabio Nobile<br />
fabio.nobile@epfl.ch<br />
École Polytechnique Fédérale de Lausanne, Station 8, CH-1015<br />
Lausanne, Switzerland<br />
Joint work with:<br />
L. Tamellini (MOX, Department of <strong>Mathematics</strong>, Politecnico di Milano, Italy),<br />
J. Beck, M. Motamed, R. Tempone (Applied <strong>Mathematics</strong> and Computational<br />
Science King Abdullah University of Science and Technology (KAUST), Kingdom of<br />
Saudi Arabia)<br />
We consider the problem of numerically approximating statistical moments of the solution<br />
of a partial differential equation (PDE), whose input data (coefficients, <strong>for</strong>cing<br />
terms, boundary conditions, geometry, etc.) are uncertain and described by a finite<br />
or countably infinite number of random variables. This situation includes the case of<br />
infinite dimensional random fields suitably expanded in e.g Karhunen-Loève or Fourier<br />
expansions.<br />
We focus on polynomial chaos approximations of the solution with respect to the<br />
underlying random variables and review common techniques to practically compute<br />
such polynomial approximation by Galerkin projection, Collocation on sparse grids or<br />
regression methods from random evaluations.<br />
We discuss in particular the proper choice of the polynomial space both <strong>for</strong> linear elliptic<br />
PDEs with random diffusion coefficient and second order hyperbolic equations with<br />
random piecewise constant wave speed. Numerical results showing the effectiveness<br />
and limitations of the approaches will be presented as well.<br />
References<br />
[1] G. Migliorati, F. Nobile, E. von Schwerin, and R. Tempone, “Analysis of the discrete<br />
L 2 projection on polynomial spaces with random evaluations,” MOX-Report 46-<br />
2011, Department of <strong>Mathematics</strong>, Politecnico di Milano, Italy, 2011. submitted.<br />
[2] M. Motamed, F. Nobile, and R. Tempone, “A stochastic collocation method <strong>for</strong><br />
the second order wave equation with a discontinuous random speed,” MOX-Report<br />
36-2011, Department of <strong>Mathematics</strong>, Politecnico di Milano, Italy, 2011. submitted.<br />
19
[3] J. Bäck, F. Nobile, L. Tamellini, and R. Tempone, “On the optimal polynomial<br />
approximation of stochastic PDEs by galerkin and collocation methods,” MOX-<br />
Report 23-2011, Department of <strong>Mathematics</strong>, Politecnico di Milano, Italy, 2011.<br />
accepted <strong>for</strong> publication on M3AS.<br />
[4] I. Babuška, F. Nobile, and R. Tempone, “A stochastic collocation method <strong>for</strong> elliptic<br />
partial differential equations with random input data,” SIAM Review, vol. 52,<br />
pp. 317–355, June 2010.<br />
20
Tuesday, 11:30–12:00<br />
Potential and Limitations of Sparse Tensor Product<br />
Stochastic Galerkin Methods<br />
Claude J. Gittelson<br />
cgittels@purdue.edu<br />
Purdue University<br />
Sparse tensor product constructions are known to be more efficient than their full<br />
tensor product counterparts in many applications. In case of low regularity, spectral<br />
approximations <strong>for</strong> random differential equations <strong>for</strong>m an exception to this rule.<br />
We consider an elliptic PDE with a random diffusion coefficient, which we assume to<br />
be expanded in a series. The solution to such an equation depends on the coefficients<br />
in this series in addition to the spatial variables. Spectral discretizations typically use<br />
tensorized polynomials to represent the parameter-dependence of the solution. Each<br />
coefficient of the solution with respect to such a polynomial basis is a function of the<br />
spatial variable, and can be approximated by finite elements.<br />
Since the importance of the polynomial basis functions can vary greatly, it is natural<br />
to expect a gain in efficiency if the coefficients are approximated in different finite element<br />
spaces. We show under what conditions such a multilevel construction reaches a<br />
higher convergence rate than approximations employing just a single spatial discretization.<br />
Sparse tensor products, which impose additional structure on the multilevel<br />
approximation, can always essentially attain the optimal convergence rate.<br />
References<br />
[1] C.J. Gittelson, Convergence rates of multilevel and sparse tensor approximations<br />
<strong>for</strong> a random elliptic PDE, Submitted.<br />
21
Tuesday, 12:00–12:30<br />
A priori error analysis of stochastic Galerkin<br />
mixed finite element methods<br />
Alex Bespalov<br />
alexey.bespalov@manchester.ac.uk<br />
School of <strong>Mathematics</strong>, University of Manchester, Ox<strong>for</strong>d Road, Manchester,<br />
M13 9PL, United Kingdom<br />
Joint work with:<br />
Catherine Powell and David Silvester<br />
(School of <strong>Mathematics</strong>, University of Manchester, United Kingdom)<br />
Stochastic Galerkin methods are becoming increasingly popular <strong>for</strong> solving PDE problems<br />
with random data (i.e., coefficients, sources, boundary conditions, geometry etc.).<br />
These methods combine conventional Galerkin finite elements on the (spatial) computational<br />
domain with spectral approximations in the stochastic variables. Whilst there<br />
exists a large body of research work on numerical approximation of primal <strong>for</strong>mulations<br />
<strong>for</strong> elliptic PDEs with random data, the case of mixed variational problems is not so<br />
well developed.<br />
In this talk we will address the issues involved in the error analysis of stochastic<br />
Galerkin approximations to the solution of a first order system of PDEs with random<br />
coefficients. First, we approximate the random input by using spectral expansions in M<br />
random variables and trans<strong>for</strong>m the variational saddle point problem to a parametric<br />
deterministic one. Approximations to the latter problem are then constructed by combining<br />
mixed hp finite elements on the computational domain with M-variate tensor<br />
product polynomials of order k = (k 1 , k 2 , . . . , k M ). We will discuss the inf-sup stability<br />
and well-posedness of the continuous and finite-dimensional problems, the regularity<br />
of solutions with respect to the M parameters describing the random coefficients, and<br />
a priori error bounds <strong>for</strong> stochastic Galerkin approximations in terms of discretisation<br />
parameters M, h, p, and k.<br />
This work is supported by the EPSRC (UK) under grant no. EP/H021205/1.<br />
22
Tuesday, 14:30–15:25<br />
A Flexible Stochastic Collocation Algorithm<br />
on Arbitrary Nodes via Interpolation<br />
Dongbin Xiu<br />
dxiu@purdue.edu<br />
Department of <strong>Mathematics</strong>, Purdue University, West Lafayette, IN 47907, USA<br />
Joint work with:<br />
Akil Narayan, Purdue University<br />
Stochastic collocation method have become the dominating methods <strong>for</strong> uncertainty<br />
quantification and stochastic computing of large and complex systems. Though the<br />
idea has been explored in the past, its popularity is largely due to the recent advance<br />
of employing high-order nodes such as sparse grids. These nodes allow one to conduct<br />
UQ simulations with high accuracy and efficiency.<br />
The critical issue is, without any doubt, the standing challenge of “curse-of-dimensionality”.<br />
For practical systems with large number of random inputs, the number of nodes <strong>for</strong><br />
stochastic collocation method can grow fast and render the method computationally<br />
prohibitive. Such kind of growth is especially severe when the nodal construction is<br />
structured, e.g., tensor grids, sparse grids, etc. One way to alleviate the difficulty is to<br />
employ adaptive approach, where the nodes are added only in the region that is needed.<br />
To this end, it is highly desirable to design stochastic collocation methods that work<br />
with arbitrary number of nodes on arbitrary locations. Another strong motivation is<br />
the practical restriction one may face. In many cases one can not conduct simulations<br />
at the desired nodes.<br />
In this work we present an algorithm that allows one to construct high-order polynomial<br />
responses based on stochastic collocation on arbitrary nodes. The method is based<br />
on constructing a “correct” polynomial space so that multi-dimensional polynomial interpolation<br />
can be constructed <strong>for</strong> any data. We present its rigorous mathematical<br />
framework, its practical implementation details, and its applications in high dimensions.<br />
23
Tuesday, 15:30–16:00<br />
Computation of maximum norm and level sets<br />
in the canonical tensor <strong>for</strong>mat with application<br />
to SPDE<br />
Alexander Litvinenko<br />
wire@tu-bs.de<br />
Technische Universität Braunschweig, Hans-Sommerstr. 65, 38106 Braunschweig<br />
Joint work with:<br />
M. Espig, W. Hackbusch (MPI <strong>for</strong> <strong>Mathematics</strong> in the Sciences, Leipzig, Germany),<br />
H. G. Matthies, E. Zander (Technische Universität Braunschweig)<br />
We introduce new methods <strong>for</strong> the analysis of high dimensional data in tensor <strong>for</strong>mats,<br />
where the underling data come from the stochastic elliptic boundary value problem.<br />
After discretisation of the deterministic operator as well as the presented random fields<br />
via KLE and PCE, the obtained high dimensional operator can be approximated via<br />
sums of elementary tensors. This tensors representation can be effectively used <strong>for</strong><br />
computing different values of interest, such as maximum norm, level sets and cumulative<br />
distribution function. The basic concept of the data analysis in high dimensions is<br />
discussed on tensors represented in the canonical <strong>for</strong>mat, however the approach can be<br />
easily used in other tensor <strong>for</strong>mats. As an intermediate step we describe efficient iterative<br />
algorithms <strong>for</strong> computing the characteristic and sign functions as well as pointwise<br />
inverse in the canonical tensor <strong>for</strong>mat. Since during majority of algebraic operations<br />
as well as during iteration steps the representation rank grows up, we use lower-rank<br />
approximation and inexact recursive iteration schemes.<br />
References<br />
[1] M. Espig, W. Hackbusch, A. Litvinenko, H. G. Matthies and E. Zander, Efficient<br />
Analysis of High Dimensional Data in Tensor Formats, Technische Universität<br />
Braunschweig, preprint 11-2011, http://www.digibib.tu-bs.de/?docid=00041268<br />
24
Tuesday, 16:00–16:30<br />
p-FEM and p-BEM – achieving high accuracy<br />
Matthias Maischak<br />
matthias.maischak@brunel.ac.uk<br />
Brunel University, Uxbridge UB8 3PH, UK<br />
It is well known, that we can achieve exponential fast convergence by approximating<br />
the solution of a pde using either the p-version fem or bem if then solution is smooth,<br />
or using the hp-version with geometrical refined mesh if the solution has singularities<br />
[2, 3]. Achieving exponential fast convergence asymptotically proves to be surprisingly<br />
difficult, due to requirements on memory, solution time and the necessary numerical<br />
precision.<br />
In this presentation we investigate changes necessary to an existing fem/bem code,<br />
software tools <strong>for</strong> converting the code to a different data type using a multi-precision<br />
package [1] and discuss implementation issues which arise from considerable more time<br />
and memory consuming arithmetic operations.<br />
We will compare standard double precision (hardware supported), quad precision (compiler<br />
supported) and higher precision (additional Software package [1]). We will present<br />
examples achieving errors with 1e-30 or less using p- and hp fem-bem methods in 1,2<br />
and 3 dimensions.<br />
References<br />
[1] David H. Bailey, Yozo Hida, Xiaoye S. Li, and Brandon Thompson, ARPREC: An<br />
Arbitrary Precision Computation Package, Technical Report LBNL-53651, 2002<br />
[2] B. Guo, and I. Babuška, The h − p version of the finite element method, part 1:<br />
The basic approximation results, Computational Mechanics, 1 (1986), 21–41.<br />
[3] N. Heuer, M. Maischak, and E.P. Stephan, Exponential convergence of the hpversion<br />
<strong>for</strong> the boundary element method on open surfaces, Numerische Mathematik,<br />
83 (1999), 641–666.<br />
25
Tuesday, 17:00–17:55<br />
To mesh or not to mesh: High order numerical<br />
simulation <strong>for</strong> complex structures<br />
Ernst Rank<br />
rank@bv.tum.de<br />
Chair <strong>for</strong> Computation in Engineering, Technische Universität München, Arcisstrasse<br />
21, 80333 München, Germany<br />
Joint work with:<br />
Stefan Kollmannsberger, Dominik Schillinger, Christian Sorger (Chair <strong>for</strong><br />
Computation in Engineering, Technische Universität München, Arcisstrasse 21, 80333<br />
München, Germany)<br />
The paper will give an overview over some recent work on high order finite element<br />
meshing <strong>for</strong> thin-walled structures as well as on a high order variant of fictitious domain<br />
methods. The basis <strong>for</strong> our research is the p-version of the finite element method.<br />
It allows <strong>for</strong> a very accurate computation of thin-walled structures using solid models<br />
and is not depending on dimensional reduction like in shell or plate models. To<br />
take full advantage of this method it is yet necessary to develop special mesh generating<br />
techniques which discretize the exact 3-dimensional curved geometry and not a<br />
facetted mid-surface of the structure. In the second part of the presentation the Finite<br />
Cell Method (FCM), an extension of high order finite element methods to a fictitious<br />
domain approach will be discussed. It completely avoids meshing, yet converges exponentially<br />
in energy norm <strong>for</strong> smooth problems. This approach turns out to be very<br />
efficient and accurate and can even be used <strong>for</strong> a real-time simulation of complex solid<br />
structures. Very recent results on an extension of the Finite Cell Method by an adaptive,<br />
hierarchic refinement allows a tight connection between geometric modeling and<br />
numerical analysis <strong>for</strong> thin-walled as well as <strong>for</strong> massive structures.<br />
26
Wednesday, 9:00–9:55<br />
Bernstein-Bézier finite elements of arbitrary order<br />
and optimal assembly procedures<br />
Mark Ainsworth<br />
m.ainsworth@strath.ac.uk<br />
University of Strathclyde<br />
Joint work with:<br />
G. Andriamaro and O. Davydov<br />
Algorithms are presented that enable the element matrices <strong>for</strong> the standard finite element<br />
space, consisting of continuous piecewise polynomials of degree n on simplicial<br />
elements in R d , to be computed in optimal complexity O(n 2 d). The algorithms (i) take<br />
account of numerical quadrature; (ii) are applicable to non-linear problems; and, (iii)<br />
do not rely on pre-computed arrays containing values of one-dimensional basis functions<br />
at quadrature points (although these can be used if desired). The elements are<br />
based on Bernstein-Bézier polynomials and are the first to achieve optimal complexity<br />
<strong>for</strong> the standard finite element spaces on simplicial elements.<br />
27
Wednesday, 10:00–10:55<br />
From h to p efficiently: balancing high or low order<br />
approximations with accuracy<br />
Spencer Sherwin<br />
s.sherwin@imperial.ac.uk<br />
Department of Aeronautics, Imperial College London, UK<br />
s.sherwin@imperial.ac.uk<br />
When approximating a PDE problem a reasonable question to ask is what is the ideal<br />
discretisation order <strong>for</strong> a desired computational error? Although we typically understand<br />
the answer in rather general terms there are comparatively little quantitive<br />
results? The issues behind this question are partly captured in the figure above where<br />
we observe the wall shear stress pattern around two small branching vessels in and<br />
anatomically realistic mode of part of the descending aorta. In the left image we observe<br />
a linear finite element approximation on a fine mesh and in the right image a<br />
coarse mesh approximation using a 6th order polynomial expansion is applied. The<br />
high order discretisation is much smoother but this accuracy typically is associated<br />
with a higher computational cost. Although the low order approximation looks more<br />
pixelated it does however capture all the salient features of the wall shear stress patterns.<br />
The spectral element and hp finite element methods methods can be considered as<br />
bridging the gap between the – traditionally low order – finite element method on one<br />
side and spectral methods on the other side. Consequently, a major challenge which<br />
arises in implementing the spectral/hp element methods is to design algorithms that<br />
per<strong>for</strong>m efficiently <strong>for</strong> both low- and high-order discretisations, as well as discretisations<br />
in the intermediate regime.<br />
In this presentation, we explain how the judicious use of different implementation<br />
strategies can be employed to achieve high efficiency across a range of polynomial<br />
28
orders [1, 2]. Further, based upon this efficient implementation, we analyse which<br />
spectral/hp discretisation minimises the computational cost to solve elliptic steady<br />
and unsteady model problems up to a predefined level of accuracy. Finally we discuss<br />
how the lessons learned from this study can be encapulated into a more generic library<br />
implementation.<br />
References<br />
[1] P.E.J. Vos, S.J. Sherwin and R.M. Kirby, From h to p efficiently: Implementing<br />
finite and spectral/hp element discretisations to achieve optimal per<strong>for</strong>mance at<br />
low and high order approximations, Journal of Computational Physics 229, (2010),<br />
5161-5181<br />
[2] P.E.J. Vos, S.J. Sherwin and R.M. Kirby, C.D. Cantwell, S.J. Sherwin, R.M. Kirby<br />
and P.H.J. Kelly, From h to p efficiently: Strategy selection <strong>for</strong> operator evaluation<br />
on hexahedral and tetrahedral elements, Computers and Fluids, to appear, 2011<br />
29
Wednesday, 11:30–12:25<br />
Domain Decomposition Preconditioning <strong>for</strong> High<br />
Order Hybrid Discontinuous Galerkin Methods<br />
on Tetrahedral Meshes<br />
Joachim Schöberl<br />
joachim.schoeberl@tuwien.ac.at<br />
Vienna University of Technology<br />
Hybrid discontinuos Galerkin methods are popular discretization methods in applications<br />
from fluid dynamics and many others. Often large scale linear systems arising<br />
from elliptic operators have to be solved. We show that standard p-version domain<br />
decomposition techniques can be applied, but we have to develop new technical tools<br />
to prove poly-logarithmic condition number estimates, in particular on tetrahedral<br />
meshes.<br />
30
Thursday, 9:00–9:55<br />
DPG method with optimal test functions.<br />
A progress report.<br />
Leszek Demkowicz<br />
leszek@ices.utexas.edu<br />
University of Texas at Austin<br />
I will present a progress report on the Discontinuous Petrov-Galerkin method proposed<br />
by Jay Gopalakrishan and myself [1, 2]. The main idea of the method is to employ<br />
(approximate) optimal test functions that are computed on the fly at the element<br />
level. If the error in approximating the optimal test functions is negligible, the method<br />
AUTOMATICALLY guarantees the discrete stability, provided the continuous problem<br />
is well posed. And this holds <strong>for</strong> ANY linear problem. The result is shocking until one<br />
realizes that we are working with an unconventional least squares method. The twist<br />
lies in the fact that the residual lives in a dual space and it is computed using dual<br />
norms.<br />
The method turns out to be especially suited <strong>for</strong> singular perturbation problems where<br />
one strives not only <strong>for</strong> stability but also <strong>for</strong> ROBUSTNESS, i.e. a stability UNIFORM<br />
with respect to the perturbation parameter. Critical to the construction of a robust<br />
version of the DPG method is the choice of the test norm that should imply the uni<strong>for</strong>m<br />
stability.<br />
I will report on recent results <strong>for</strong> two important model problems: convection-dominated<br />
diffusion and linear acoustics equations [3, 4, 5]. I will attempt to outline a general<br />
strategy <strong>for</strong> selecting the optimal test norm and will focus on the task of approximating<br />
the optimal test functions and effects of such an approximation [6].<br />
References<br />
[1] L. Demkowicz and J. Gopalakrishnan. A Class of Discontinuous Petrov-Galerkin<br />
Methods. Part II: Optimal Test Functions. Numer. Meth. Part. D. E., 27, 70-105,<br />
2011.<br />
[2] L. Demkowicz, J. Gopalakrishnan and A. Niemi, A Class of Discontinuous Petrov-<br />
Galerkin Methods. Part III: Adaptivity, ICES Report 2010-01, App. Num Math.,<br />
in print.<br />
[3] L. Demkowicz, J. Gopalakrishnan, I. Muga, and J. Zitelli. Wavenumber Explicit<br />
Analysis <strong>for</strong> a DPG Method <strong>for</strong> the Multidimensional Helmholtz Equation”, ICES<br />
Report 2011-24, CMAME, in print.<br />
31
[4] L. Demkowicz, N. Heuer, Robust DPG Method <strong>for</strong> Convection-Dominated Diffusion<br />
Problems. ICES Report 2011-33, submitted to SIAM J. Num. Anal.<br />
[5] L. Demkowicz, J. Gopalakrishnan, I. and Muga, D. Pardo, A Pollution Free DPG<br />
Method <strong>for</strong> Multidimensional Helmholtz Equation, in preparation.<br />
[6] J. Gopalakrishnan and W. Qiu. An Analysis of the Practical DPG Method. IMA<br />
Report 2011-2374.<br />
32
Thursday, 10:00–10:55<br />
Application of hp–Adaptive Discontinuous Galerkin<br />
Methods to Bifurcation Phenomena in Pipe Flows<br />
Paul Houston<br />
Paul.Houston@nottingham.ac.uk<br />
School of Mathematical Sciences, University of Nottingham,<br />
Nottingham NG7 2RD, UK<br />
In this talk we consider the a posteriori error estimation and hp–adaptive mesh refinement<br />
of discontinuous Galerkin finite element approximations of the bifurcation<br />
problem associated with the steady incompressible Navier–Stokes equations. Particular<br />
attention is given to the reliable error estimation of the critical Reynolds number<br />
at which a steady pitch<strong>for</strong>k bifurcation occurs when the underlying physical system<br />
possesses rotational and reflectional or O(2) symmetry. <strong>Here</strong>, computable a posteriori<br />
error bounds are derived based on employing the generalization of the standard<br />
Dual Weighted Residual approach, originally developed <strong>for</strong> the estimation of target<br />
functionals of the solution, to bifurcation problems; see [1, 2, 3] <strong>for</strong> details. Numerical<br />
experiments highlighting the practical per<strong>for</strong>mance of the proposed a posteriori error<br />
indicator on h– and hp–adaptively refined computational meshes are presented. <strong>Here</strong>,<br />
particular attention is devoted to the problem of flow through a cylindrical pipe with<br />
a sudden expansion, which represents a notoriously difficult computational problem.<br />
This research has been carried out in collaboration with Andrew Cliffe and Edward Hall<br />
(University of Nottingham), Tom Mullin and James Seddon (University of Manchester),<br />
and Eric Phipps and Andy Salinger (Sandia National Laboratories).<br />
References<br />
[1] K.A. Cliffe, E. Hall, P. Houston, E.T. Phipps, and A.G. Salinger. Adaptivity and<br />
A Posteriori Error Control <strong>for</strong> Bifurcation Problems I: The Bratu Problem. Communications<br />
in Computational Physics 8(4):845-865, 2010.<br />
[2] K.A. Cliffe, E. Hall, P. Houston, E.T. Phipps, and A.G. Salinger. Adaptivity and A<br />
Posteriori Error Control <strong>for</strong> Bifurcation Problems II: Incompressible fluid flow in<br />
Open Systems with Z 2 Symmetry. Journal of Scientific Computing 47(3):389-418,<br />
2011.<br />
[3] K.A. Cliffe, E. Hall, P. Houston, E.T. Phipps, and A.G. Salinger. Adaptivity and<br />
A Posteriori Error Control <strong>for</strong> Bifurcation Problems III: Incompressible fluid flow<br />
in Open Systems with O(2) Symmetry. Journal of Scientific Computing (in press).<br />
33
Thursday, 11:30–12:25<br />
Exponential convergence of hp-version<br />
discontinuous Galerkin methods <strong>for</strong> elliptic<br />
problems in polyhedral domains<br />
Dominik Schötzau<br />
schoetzau@math.ubc.ca<br />
<strong>Mathematics</strong> Department, University of British Columbia, Vancouver, BC V6T 1Z2,<br />
Canada<br />
Joint work with:<br />
Christoph Schwab (Seminar of Applied <strong>Mathematics</strong>, ETH Zürich, 8092 Zürich,<br />
Switzerland, email: schwab@math.ethz.ch),<br />
Thomas Wihler (Mathematisches Institut, Universität Bern, 3013 Bern, Switzerland,<br />
email: wihler@math.unibe.ch)<br />
We design and analyze hp-version interior penalty (IP) discontinuous Galerkin (dG)<br />
finite element methods <strong>for</strong> the numerical approximation of linear second order elliptic<br />
boundary-value problems in three dimensional polyhedral domains. The methods are<br />
based on using hexahedral meshes that are geometrically and anisotropically refined<br />
towards corners and edges. Similarly, the local polynomial degrees are increased linearly<br />
and possibly anisotropically away from singularities. We prove that the methods are<br />
well-defined <strong>for</strong> problems with singular solutions and are stable under the proposed<br />
hp-refinements. Finally, we show that the hp-dG methods lead to exponential rates of<br />
convergence in the number of degrees of freedom <strong>for</strong> problems with piecewise analytic<br />
data.<br />
References<br />
[1] D. Schötzau, C. Schwab, and T. Wihler: hp-dGFEM <strong>for</strong> second order elliptic problems<br />
in polyhedra I: Stability and quasioptimality on geometric meshes, submitted.<br />
[2] D. Schötzau, C. Schwab, and T. Wihler: hp-dGFEM <strong>for</strong> second order elliptic<br />
problems in polyhedra II: Exponential convergence, submitted.<br />
34
Thursday, 14:30–15:25<br />
A Posteriori Error Analysis <strong>for</strong> Linear Parabolic<br />
PDE based on hp-discontinuous Galerkin<br />
Time-Stepping<br />
Thomas P. Wihler<br />
wihler@math.unibe.ch<br />
<strong>Mathematics</strong> Institute, University of Bern, Switzerland<br />
Joint work with:<br />
Omar Lakkis (U of Sussex, UK),<br />
Manolis Georgoulis (U of Leicester, UK)<br />
Dominik Schötzau (UBC), Canada<br />
We consider full discretizations in time and space of linear parabolic PDE. More precisely,<br />
in order to discretize the time variable the hp-discontinuous Galerkin time stepping<br />
scheme from [3] is employed. Furthermore, con<strong>for</strong>ming variational methods are<br />
applied to approximate the problem in space.<br />
The focus of the talk is the development of an L 2 (H 1 )-based a posteriori error analysis<br />
(in time and space) <strong>for</strong> the given class of schemes. <strong>Here</strong>, the key point is the combination<br />
of an hp-time reconstruction (see [4], cf. also [2]) and of an elliptic reconstruction<br />
in space (see [1]). The resulting a posteriori error estimate is expressed in terms of the<br />
fully discrete solution.<br />
References<br />
[1] C. Makridakis and R. H. Nochetto, Elliptic reconstruction and a posteriori error<br />
estimates <strong>for</strong> parabolic problems, SIAM Journal on Numerical Analysis, 41 (2003),<br />
No. 4, 1585–1594.<br />
[2] C. Makridakis and R.H. Nochetto, A posteriori error analysis <strong>for</strong> higher order<br />
dissipative methods <strong>for</strong> parabolic problems, Numerische Mathematik, 104 (2006),<br />
489–514.<br />
[3] D. Schötzau and C. Schwab, Time discretization of parabolic problems by the hpversion<br />
of the discontinuous Galerkin finite element method, SIAM Journal on<br />
Numerical Analysis, 38 (2000), 837–875.<br />
[4] D. Schötzau and T. P. Wihler, A posteriori error estimation <strong>for</strong> hp-version timestepping<br />
methods <strong>for</strong> parabolic partial differential equations, Numerische Mathematik,<br />
115 (2010), no. 3, 475–509.<br />
35
Thursday, 15:30–16:25<br />
Certified Reduced Basis Methods <strong>for</strong> Integral<br />
Equations with Applications to Electromagnetics<br />
Jan S Hesthaven<br />
Jan.Hesthaven@Brown.edu<br />
Division of Applied <strong>Mathematics</strong>, Brown University, Providence, RI 02912, USA<br />
Joint work with:<br />
Yvon Maday (Laboratory of Jacques-Louis Lions, University of Paris VI,<br />
Paris, France),<br />
Benjamin Stamm (Department of <strong>Mathematics</strong>, UC Berkeley, CA 94720, USA),<br />
Shun Zhang (Division of Applied <strong>Mathematics</strong>, Brown University,<br />
Providence, RI 02912, USA)<br />
The development and application of models of reduced computational complexity is<br />
used extensively throughout science and engineering to enable the fast/real-time/subscale<br />
modeling of complex systems <strong>for</strong> control, design, prediction purposes of multi-scale<br />
problems, uncertainty quantification etc. Such models, while often successful and of<br />
undisputed value, are, however, often heuristic in nature and the validity and accuracy<br />
of the output is often unknown. This limits the predictive value of these models.<br />
In this talk we shall focus on parametrized models based on integral equations. These<br />
are used widely to model a variety of important problems in acoustics, heat conduction,<br />
and electromagnetics and the analysis and computational techniques have been well<br />
developed during the last decades. However, <strong>for</strong> many-query situations the direct<br />
solution of these models may be prohibitive and the development of reduced models<br />
of significant value. In the context of integral equations <strong>for</strong> electromagnetic scattering,<br />
variations over frequency, sources, observations etc, can be substantial, yet a rapid<br />
evaluation has important application in design and detection applications.<br />
We discuss the development of certified reduced basis methods <strong>for</strong> integral equations to<br />
enable a rapid and error-controlled evaluation of the scattering over parametric variation.<br />
During this talk we outline the basic elements of a certified reduced basis method<br />
and highlight specific issues related to integral equations caused by their non-affine<br />
nature and the need to attend to computational complexity due to large parameter<br />
variations.<br />
We continue by discussing how these reduced models can be combined iteratively to<br />
solve larger multi-object problems without directly assembling the full physical problem.<br />
Time permitting we extend this discussion to the high-dimensional parametrized<br />
problem and discuss ways of achieving accurate parametric compression in the reduced<br />
model using ANOVA expansions.<br />
36
Thursday, 17:00–17:55<br />
Convergence of hp-BEM <strong>for</strong> the electric field<br />
integral equation<br />
Ralf Hiptmair<br />
hiptmair@sam.math.ethz.ch<br />
ETH Züich<br />
Joint work with:<br />
A. Bespalov (University of Manchester),<br />
N. Heuer (Pontificia Universidad Católica de Chile, Santiago)<br />
We consider the variational <strong>for</strong>mulation of the electric field integral equation (EFIE)<br />
on bounded polyhedral open or closed surfaces. We employ a con<strong>for</strong>ming Galerkin<br />
discretization based on div Γ -con<strong>for</strong>ming Raviart-Thomas boundary elements (BEM)<br />
of locally variable polynomial degree on shape-regular surface meshes. We establish<br />
asymptotic quasi-optimality of Galerkin solutions on sufficiently fine meshes or <strong>for</strong><br />
sufficiently high polynomial degree.<br />
References<br />
[1] A. Bespalov, N. Heuer, and R. Hiptmair, Convergence of natural hp-BEM <strong>for</strong> the<br />
electric field integral equation on polyhedral surfaces,, SIAM J. Numer. Anal., 48<br />
(2010), pp. 1518–1529.<br />
37
Friday, 9:00–9:55<br />
On local refinement in isogeometric analysis<br />
Annalisa Buffa<br />
annalisa@imati.cnr.it<br />
Istituto di Matematica Applicata e Tecnologie In<strong>for</strong>matiche ’E. Magenes’, CNR,<br />
Via Ferrata 1, Pavia, Italy<br />
Isogeometric methodologies are designed with the aim of improving the connection<br />
between numerical simulation of physical phenomena and the Computer Aided Design<br />
systems. Indeed, the ultimate goal is to eliminate or drastically reduce the approximation<br />
of the computational domain and the re-meshing by the use of the “exact”<br />
geometry directly on the coarsest level of discretization. This is achieved by using<br />
B-Splines or Non Uni<strong>for</strong>m Rational B-Splines <strong>for</strong> the geometry description as well as<br />
<strong>for</strong> the representation of the unknown fields. The use of Spline or NURBS functions,<br />
together with isoparametric concepts, results in an extremely successfully idea and<br />
paves the way to many new numerical schemes enjoying features that would be extremely<br />
hard to achieve within a standard finite element framework. One of the big<br />
challenge in the context of isogeometric analysis is the possibility to break the tensor<br />
product structure of Splines and so design adaptive methods. T-splines, introduced<br />
by T. Sederberg et al. in 2004, have been used to this aim in geometric modeling and<br />
design, but their use in isogeometric analysis requires a lot of care.<br />
In this talk, after a short introduction about Isogeometric analysis, I will report on our<br />
experience as regards the use of T-splines as a tool <strong>for</strong> local refinement.<br />
References<br />
[1] L. Beirao da Veiga, A. Buffa, D. Cho, G. Sangalli, Analysis-Suitable T-splines are<br />
Dual-Compatible. Tech. Rep. IMATI CNR, 2012.<br />
[2] A. Buffa, D. Cho, M. Kumar, Characterization of T-splines with reduced continuity<br />
order on T-meshes, Comput. Methods Appl. Mech. Engrg., Volumes 201204, 1<br />
January 2012, pp. 112-126.<br />
[3] L Beirao da Veiga, A. Buffa, D. Cho, G. Sangalli, Isogeometric analysis using<br />
T-splines on two-patch geometries, Computer Methods in Applied Mechanics and<br />
Engineering, Vol. 200 (21-22), pp.1787-1803, (2011)<br />
38
Friday, 10:00–10:30<br />
Domain decomposition methods<br />
in Isogeometric analysis<br />
Lourenco Beirão da Veiga<br />
lourenco.beirao@unimi.it<br />
Department of <strong>Mathematics</strong>, University of Milan,<br />
Via Saldini 50, 20133, Milan, Italy.<br />
Joint work with:<br />
Durkbin Cho (University of Milan),<br />
Luca Pavarino (University of Milan),<br />
Simone Scacchi (University of Milan)<br />
NURBS-based isogeometric analysis (IGA in short) [3] is a recent and innovative numerical<br />
methodology <strong>for</strong> the analysis of PDE problems, that adopts NURBS basis<br />
functions <strong>for</strong> the description of the discrete space and allows <strong>for</strong> an exact description<br />
of CAD-type geometries.<br />
The discrete systems produced by isogeometric methods are better conditioned than<br />
the systems produced by standard finite elements or finite differences, but their conditioning<br />
can still degenerates rapidly <strong>for</strong> decreasing mesh size h, increasing polynomial<br />
degrees p and increasing coefficient jumps in the elliptic operator. There<strong>for</strong>e the design<br />
and analysis of efficient iterative solvers <strong>for</strong> isogeometric analysis is a challenging<br />
research topic, particularly <strong>for</strong> three-dimensional problems with discontinuous coefficients.<br />
The global high regularity of the adopted NURBS functions introduces a new<br />
set of difficulties to be dealt with, both at the practical and the theoretical level.<br />
In the present talk we will focus on Domain Decomposition [4] methods <strong>for</strong> Isogeometric<br />
Analysis concentrating our ef<strong>for</strong>ts on a model elliptic problem. We will consider both<br />
an Overlapping Schwarz [1] method and a BDDC (i.e. non overlapping) method [2]. In<br />
both cases we will show theoretical condition number bounds in terms of the mesh sizes<br />
h, H (such as the scalability property) and numerical tests that confirm and extend in<br />
terms of the polynomial order p and regularity index k the theoretical predictions.<br />
The theoretical and numerical results of the present talk are strongly encouraging and<br />
show that domain decomposition seems to be a very good choice <strong>for</strong> preconditioning<br />
and solving problems in Isogeometric Analysis. Clearly more work needs to be accomplished,<br />
both in terms of extension to more complex problems and of development (and<br />
comparison with) different preconditioning and solving schemes.<br />
39
References<br />
[1] L. Beirao da Veiga, D. Cho, L. F. Pavarino, S. Scacchi, Overlapping Schwarz<br />
Methods <strong>for</strong> Isogeometric Analysis, to appear on Siam J. Numer. Anal. and Preprint<br />
IMATI-CNR (2011).<br />
[2] L. Beirao da Veiga, D. Cho, L. F. Pavarino, S. Scacchi, BDDC preconditioners <strong>for</strong><br />
Isogeometric Analysis, submitted <strong>for</strong> publication and Preprint IMATI-CNR (2012).<br />
[3] J. A. Cottrell, T. J. R. Hughes, Y. Basilevs, Isogeometric Analysis: Toward Integration<br />
of CAD and FEA, Wiley, Chichester, U.K., 2009<br />
[4] A. Toselli, O. B. Widlund, Domain Decomposition Methods: Algorithms and Theory,<br />
Computational <strong>Mathematics</strong>, Vol. 34. Springer-Verlag, Berlin, 2004.<br />
40
Friday, 10:30–11:00<br />
Isogeometric Collocation Methods <strong>for</strong> Elasticity<br />
Alessandro Reali<br />
alessandro.reali@unipv.it<br />
Structural Mechanics Department, University of Pavia<br />
Joint work with:<br />
Ferdinando Auricchio (University of Pavia),<br />
Lourenco Beirão da Veiga (University of Milan),<br />
Thomas J.R. Hughes (University of Texas at Austin),<br />
Giancarlo Sangalli(University of Pavia)<br />
Isogeometric Analysis (IGA) is a recent idea, firstly introduced by Hughes et al. [1, 2] to<br />
bridge the gap between Computational Mechanics and Computer Aided Design (CAD).<br />
The key feature of IGA is to extend the finite element method representing geometry by<br />
functions, such as Non-Uni<strong>for</strong>m Rational B-Splines (NURBS), which are used by CAD<br />
systems, and then invoking the isoparametric concept to define field variables. Thus,<br />
the computational domain exactly reproduces the NURBS description of the physical<br />
domain. Moreover, numerical testing in different situations has shown that IGA holds<br />
great promises, with a substantial increase in the accuracy-to-computational-ef<strong>for</strong>t ratio<br />
with respect to standard finite elements, also thanks to the high regularity properties<br />
of the employed functions.<br />
In the framework of NURBS-based IGA, collocation methods have been recently proposed<br />
by Auricchio et al. [3], constituting an interesting high-order low-cost alternative<br />
to standard Galerkin approaches. Such techniques have also been successfully applied<br />
to elastostatics and explicit elasodynamics [4].<br />
In this work, after an introduction to isogeometric collocation methods, we move to the<br />
solution of elasticity problems and present in detail the results discussed in [4]. Particular<br />
attention is devoted to the imposition of boundary conditions (of both Dirichlet<br />
and Neumann type) and to the treatment of the multi-patch case. Also the development<br />
of explicit high-order (in space) collocation methods <strong>for</strong> elastodynamics is here<br />
considered and studied. Several numerical experiments are presented in order to show<br />
the good behavior of these approximation techniques.<br />
References<br />
[1] Hughes, T.J.R., Cottrell, J.A., Bazilevs, Y., Isogeometric analysis: CAD, finite<br />
elements, NURBS, exact geometry, and mesh refinement, Computer Methods in<br />
Applied Mechanics and Engineering, 194, 4135-4195 (2005).<br />
41
[2] Cottrell, J.A., Hughes, T.J.R., Bazilevs, Y., Isogeometric Analysis. Towards integration<br />
of CAD and FEA. Wiley, (2009).<br />
[3] Auricchio, F., Beiro da Veiga, L., Hughes, T.J.R., Reali, A., Sangalli, G., Isogeometric<br />
Collocation Methods, Mathematical Models and Methods in Applied Sciences,<br />
20, 2075-2107 (2010).<br />
[4] Auricchio, F., Beiro da Veiga, L., Hughes, T.J.R., Reali, A., Sangalli, G., Isogeometric<br />
collocation <strong>for</strong> elastostatics and explicit dynamics, ICES Report (2012).<br />
42
Friday, 11:30–12:25<br />
Isogeometric discretizations of the Stokes problem<br />
Giancarlo Sangalli<br />
giancarlo.sangalli@unipv.it<br />
Dipartimento di Matematica, Università di Pavia<br />
and IMATI-CNR “Enrico Magenes”, Pavia<br />
Joint work with:<br />
Andrea Bressan (Dipartimento di Matematica, Università di Pavia),<br />
Annalisa Buffa (IMATI-CNR “Enrico Magenes”, Pavia),<br />
Carlo de Falco (MOX, Milano),<br />
Rafael Vázquez (IMATI-CNR “Enrico Magenes”, Pavia)<br />
Isogeometric analysis has been introduced in 2005 by T.J.R. Hughes and co-authors<br />
as a novel technique <strong>for</strong> the discretization of Partial Differential Equations (PDE).<br />
This technique is having a growing impact on several fields, from fluid dynamics, to<br />
structural mechanics, and electromagnetics. A comprehensive reference is the book<br />
[6]. Isogeometric methodologies adopt B-Splines or Non-Uni<strong>for</strong>m Rational B-Splines<br />
(NURBS) functions <strong>for</strong> the geometry description as well as <strong>for</strong> the representation of<br />
the unknown fields. Splines and NURBS offer a flexible set of basis functions <strong>for</strong><br />
which mesh refinement and degree elevation are very efficient. Beside the fact that<br />
in isogeometric analysis one can directly treat geometries described by Splines and<br />
NURBS parametrizations, these functions are interesting in themselves since they easily<br />
allow global smoothness beyond the classical C 0 -continuity of FEM.<br />
This talk presents the isogeometric methods <strong>for</strong> the Stokes studied in [2, 3].<br />
In [2] we have analyzed discretizations that extend the Taylor-Hood element, wellknown<br />
in the finite element context, to the isogeometric context. These isogeometric<br />
Taylor-Hood elements are based on p + 1 degree NURBS velocity approximation and p<br />
degree NURBS pressure approximation. The novelty, in the isogeometric framework, is<br />
that C r global regularity is allowed, up to r = p−1. The stability analysis is per<strong>for</strong>med.<br />
In [3] we have proposed a smooth Raviart-Thomas isogeometric discretization that<br />
provides an exact divergence-free discrete solution. This is possible thanks to the<br />
commuting de Rham digram property that this element fulfils (see [5, 4]).<br />
References<br />
[1] Andrea Bressan. Isogeometric regular discretization <strong>for</strong> the Stokes problem. IMA<br />
Journal of Numerical Analysis, 2010.<br />
43
[2] Andrea Bressan, Giancarlo Sangalli Isogeometric discretizations of the Stokes problem:<br />
stability analysis by the macroelement technique. IMA Journal of Numerical<br />
Analysis, submitted.<br />
[3] A. Buffa, C. de Falco, and G. Sangalli. Isogeometric analysis: stable elements <strong>for</strong><br />
the 2d stokes equation. International Journal <strong>for</strong> Numerical Methods in Fluids,<br />
65(11-12):1407–1422, 2011.<br />
[4] A. Buffa, J. Rivas, G. Sangalli, and R. Vázquez. Isogeometric discrete differential<br />
<strong>for</strong>ms in three dimensions. SIAM Journal on Numerical Analysis, 49:818, 2011.<br />
[5] A. Buffa, G. Sangalli, and R. Vázquez. Isogeometric analysis in electromagnetics:<br />
B-splines approximation. Computer Methods in Applied Mechanics and Engineering,<br />
199(17-20):1143–1152, 2010.<br />
[6] J.A. Cottrell, T.J.R. Hughes, and Y. Bazilevs. Isogeometric analysis: toward integration<br />
of CAD and FEA. John Wiley & Sons Inc, 2009.<br />
44
Friday, 14:30–15:00<br />
Numerical homogenization procedures in solid<br />
mechanics using the finite cell method<br />
Alexander Düster<br />
alexander.duester@tu-harburg.de<br />
Numerische Strukturanalyse mit Anwendungen in der Schiffstechnik (M-10)<br />
TU Hamburg-Harburg<br />
Joint work with:<br />
H.-G. Sehlhorst, M. Joulaian (TU Hamburg-Harburg), E. Rank (TU München)<br />
A numerical procedure <strong>for</strong> the homogenization of heterogeneous and foamed materials<br />
is presented. Based on this approach, effective material properties are determined<br />
which can be utilized in finite element computations of engineering structures. The<br />
numerical homogenization strategy is applied to model and compute three-dimensional<br />
composites and sandwich structures which are composed of a foamed core covered by<br />
two faceplates. The homogenization is per<strong>for</strong>med by applying the Finite Cell Method<br />
(FCM) which can be interpreted as a combination of a fictitious domain method with<br />
high-order finite elements. Applying the FCM dramatically simplifies the meshing process,<br />
since a simple Cartesian grid is used and the geometry of the microstructure and<br />
the different material properties are taken care of during the integration of the stiffness<br />
matrices of the cells. In this way, the ef<strong>for</strong>t is shifted from mesh generation towards the<br />
numerical integration, which can be per<strong>for</strong>med adaptively in a fully automatic way. It<br />
will be demonstrated by several numerical examples that this approach can be applied<br />
to analyze a computer-aided design of a new heterogeneous material or data obtained<br />
by a three-dimensional computer-tomography of the material of interest.<br />
References<br />
[1] A. Düster, H.-G. Sehlhorst, and E. Rank, Numerical homogenization of heterogeneous<br />
and cellular materials utilizing the Finite Cell Method, Computational<br />
Mechanics, (2012), DOI: 10.1007/s00466-012-0681-2.<br />
[2] A. Düster, J. Parvizian, Z. Yang, and E. Rank, The Finite Cell Method <strong>for</strong> 3D<br />
problems of solid mechanics, Computer Methods in Applied Mechanics and Engineering,<br />
(2008), 197:3768–3782.<br />
[3] J. Parvizian, A. Düster, and E. Rank, Finite Cell Method: h- and p-extension <strong>for</strong><br />
embedded domain problems in Solid Mechanics, Computational Mechanics, (2007),<br />
41:121-133.<br />
45
Friday, 15:00–15:30<br />
Simulating the mechanical response of arteries<br />
by p-FEMs<br />
Zohar Yosibash<br />
zohary@bgu.ac.il<br />
Head-Computational Mechanics Lab, Dept. of Mechanical Engineering,<br />
Ben-Gurion University of the Negev, Beer-Sheva, Israel<br />
Joint work with:<br />
Elad Priel (Dept. of Mechanical Engineering, Ben-Gurion University of the Negev,<br />
Beer-Sheva, Israel)<br />
The healthy human artery wall is a complex biological structure whose mechanical response<br />
is of major interest and attracted a significant amount of research, mainly related<br />
to its passive response. <strong>Here</strong>in we present a high-order FE method <strong>for</strong> the treatment<br />
of the highly non-linear, almost-incopressible hyper-elastic constitutive model <strong>for</strong> the<br />
coupled passive-active response based on a transversely isotropic strain-energy-densityfunction<br />
(SEDF).<br />
More specifically, we first address SEDF of the passive response [1]. The new p-FE algorithms<br />
developed to expedite the numerical computations are presented. Numerical<br />
examples are provided to demonstrate the efficiency of the p-FEMs compared to classical<br />
h-FEMs, and thereafter the influence of the slight compressibility on the results.<br />
Although the active response (when smooth muscle cells are activated) has a significant<br />
influence on the overall mechanical response of the artery wall, it has been scarcely<br />
investigated because of the difficulty of representing such an influence in a constitutive<br />
model and lack of experimental evidence that can validate such models. This active<br />
response is strongly coupled with the passive response because of the mutual interaction<br />
(smooth muscle cells react as a function of the mechanical response). We incorporated<br />
the coupled passive-active response in our p-FE simulations that will also presented<br />
[2].<br />
References<br />
[1] Zohar Yosibash and Elad Priel, p-FEMs <strong>for</strong> hyperelastic anisotropic nearly incompressible<br />
materials under finite de<strong>for</strong>mations with applications to arteries simulation,<br />
International Journal <strong>for</strong> Numerical Methods in Engineering, 88, (2011),<br />
1152–1174.<br />
[2] Zohar Yosibash and Elad Priel, Artery active mechanical response: High order<br />
finite element implementation and investigation, Submitted <strong>for</strong> publication, (2012)<br />
46
Friday, 16:00–16:55<br />
A generalized empirical interpolation method<br />
based on moments and application<br />
Yvon Maday<br />
maday@ann.jussieu.fr<br />
UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris,<br />
France<br />
Joint work with:<br />
Olga Mula (UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions,<br />
F-75005, Paris and CEA Saclay, DEN/DANS/DM2S/SERMA/LENR, France)<br />
The empirical interpolation method introduced in [1] (see also [2]) allows to provide a<br />
simple and efficient construction of an interpolation process on manifold of small Kolmogorov<br />
width. This approach has attracted some attention in the frame of the reduced<br />
basis method and more generally reduced order approximation <strong>for</strong> the approximation<br />
of nonlinear PDE’s. The application range is nevertheless much larger.<br />
We propose in this talk a generalization of this concept with applications to combined<br />
assimilation-simulation technics <strong>for</strong> complex phenomenon.<br />
References<br />
[1] Barrault, Maxime and Maday, Yvon and Nguyen, Ngoc Cuong and Patera, Anthony<br />
T. An ‘empirical interpolation’ method: application to efficient reduced-basis<br />
discretization of partial differential equations, Comptes Rendus Mathématique.<br />
Académie des Sciences. Paris, 339 (2004) 9, pp 667–672<br />
[2] Maday, Yvon and Nguyen, Ngoc Cuong and Patera, Anthony T. and Pau, George<br />
S. H. A general multipurpose interpolation procedure: the magic points, Commun.<br />
Pure Appl. Anal., 8, (2009), 1, pp 383–404.<br />
47
List of Participants<br />
Ainsworth Mark University of Strathclyde<br />
Andreev Roman ETH Zürich<br />
Banz Lothar Leibniz Universität Hannover<br />
Bartha Ferenc University of Bergen<br />
Beirão da Veiga Lourenco Università di Milano<br />
Bespalov Alexey University of Manchester<br />
Beuchler Sven Universität <strong>Bonn</strong><br />
Bierig Claudio Universität <strong>Bonn</strong><br />
Brandsmeier Holger ETH Zürich<br />
Buffa Annalisa CNR, Pavia<br />
Canuto Claudio Politecnico di Torino<br />
Chernov Alexey Universität <strong>Bonn</strong><br />
Costabel Martin Université de Rennes 1<br />
Dauge Monique Université de Rennes 1<br />
Demkowicz Leszek University of Texas at Austin<br />
Düster Alexander TU Hamburg-Harburg<br />
Egger Herbert TU München<br />
Gittelson Claude Jeffrey Purdue University<br />
Hackbusch Wolfgang MPI Leipzig<br />
Hakula Harri Aalto University School of Science and Technology<br />
Hesthaven Jan S. Brown University<br />
Heuveline Vincent KIT, Karlsruhe<br />
Hiptmair Ralf ETH Zürich<br />
Houston Paul University of Nottingham<br />
Klose Roland CST AG, Darmstadt<br />
Litvinenko Alexander TU Braunschweig<br />
Maday Yvon Université Pierre et Marie Curie<br />
Maischak Matthias Brunel University, Uxbridge<br />
Melenk Jens Markus TU Wien<br />
Merabet Ismail Université de Valenciennes<br />
Nicaise Serge Université de Valenciennes<br />
Nobile Fabio EPFL Lausanne<br />
Pham Duong Thanh Universität <strong>Bonn</strong><br />
Quarteroni Alfio EPFL Lausanne<br />
Rank Ernst TU München<br />
Reali Alessandro Università di Pavia<br />
Reinarz Anne Universität <strong>Bonn</strong><br />
48
Sangalli Giancarlo Università di Pavia<br />
Schmidt Kersten TU Berlin<br />
Schöberl Joachim TU Wien<br />
Schötzau Dominik University of British Columbia, Vancouver<br />
Schröder Andreas HU Berlin<br />
Schwab Christoph ETH Zürich<br />
Sherwin Spencer J. Imperial College London<br />
Sloan Ian H. UNSW Sydney<br />
Sprungk Bjoern TU Bergakademie Freiberg<br />
Stamm Benjamin University of Cali<strong>for</strong>nia, Berkeley<br />
Stephan Ernst P. Leibniz Universität Hannover<br />
Valli Alberto Università di Trento<br />
Weggler Lucy Universität des Saarlandes<br />
Wihler Thomas Universität Bern<br />
Xiu Dongbin Purdue University<br />
Yosibash Zohar Ben-Gurion University, Beer-Sheva<br />
Zaglmayr Sabine CST AG, München<br />
49