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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
9:00Stephan Sloan Ainsworth Demkowicz Buffa
10:00Canuto Nobile Sherwin Houston
Beirao
da Veiga10:30
Reali
11:00 Coffee break
11:30Quarteroni Gittelson Schoberl Schotzau Sangalli
12:00Bespalov
12:30 Lunch break
14:30Dauge Xiu Wihler Duster
Excursions:15:00
YosibashArithmeum
15:30Melenk Litvinenko Hesthaven
orCoffee break
16:00Maischak Maday
Botanic16:30 Coffee break
GardensCoffee break
17:00Hackbusch Rank Hiptmair
–18:00Free Time
20:00 Dinner
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High-Order Numerical Approximation
for Partial Differential Equations
Over the last decades high-order numerical approximation for PDEs have become well-established tools for the efficient, highly accurate numerical solution of (boundary)integral equations and partial differential equations in computational science and engi-neering; prominent examples are hp- and spectral element methods in application areassuch as computational fluid dynamics, computational climate modelling, to name buta few. Their theoretical convergence analysis and adaptive hp- and spectral elementversions currently experience strong development. New application areas for spectralmethods are infinite dimensional Fokker-Planck equations, stochastic partial differen-tial equations, uncertainty quantification and isogeometric analysis. Here, analysis ismissing to a large extent and is subject of current research.
The aim of the Workshop is to bring together leading experts in analysis and imple-mentation of High-Order Methods for PDEs, thereby stimulating an intensive ideaexchange 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 touse Rooms 1.007 and 1.008 as well as the blackboard in the Plucker Hall (Room 1.015)for discussions. Please feel free to use the mathematics library in the ground floor. Ithas a large collection of books and journals.
DiscussionRoom1.008
DiscussionRoom1.007
Plucker Hall1.015
(coffeebreaks)
Lipschitz Hall1.016
(lecture room)
Stairs down
(Library /Ground floor)
MainEntrance(1st Floor)
Elevator
Stairs up
Endenicher Alle
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Getting there
The most convenient way to get to the workshop venue from the hotel “My Poppels-dorf” is walking (about 10-15 min). Alternatively, you might take a bus 631 fromthe stop “Poppelsdorfer Platz” to the stop “Kaufmannstrasse” (4 stops, goes every 30min.) or a taxi.
Uni-Hauptgebäude
Opernhaus
Rat-haus
Post
Stadthaus
PoppelsdorferSchloss
Hofgarten
AlterFriedhof
BotanischeGärten
Wallfahrtsweg 4Hotel „My Poppelsdorf”
Bonngasse 30Restaurant „Im Stiefel”
Endenicher Allee 60Workshop Venue
Meckenheimer Allee 171Botanic Gardens
Lennéstr. 2Arithmeum
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Bonngasse
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Beethovenstr.
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Lunch and coffee breaks
Coffee will be served during the coffee breaks in the Plucker Hall (Room 1.015), next tothe Lipschitz Hall where all talks will take place. For the lunch you might go to a placeof your choice. Many restaurants are located in the Clemens-August street (you shouldhave 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 MathematicsCenter, cross the street and turn to the right).
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Internet access via WLAN
We offer a free Internet access via WLAN in the Workshop Venue. For this, you needan 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 deskor 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 historicalmechanical 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 willbe expected at 15:00 at the entrance to Arithmeum (Lenne 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 thenumber of participants) applies for the excursion in the Botanic Gardens. We kindlyask you to inform us during the registration, or by Monday evening the latest, whichexcursion would you like to join.
Dinner on Thursday
On Thursday’s evening we welcome you to join us about 20:00 for the Workshop Dinnerat the “typical” restaurant Im Stiefel in the center of Bonn (Bonngasse 30). Unfor-tunately, 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) rangebetween 8 and 14 Euro.
We kindly ask you to inform us during the registration, or by Monday evening thelatest, if you wish to join us for the Workshop Dinner.
Acknowledgement
The organizers would like to thank the Hausdorff Center for Mathematics for thegenerous financial support, which made this Workshop possible.
Special thanks go to Laura Siklossy and other members of the HIM and HCM Admin-istration & Support teams for the strong an continuous support in all administrativeissues. Many thanks to Gunder-Lily Sievert and her team for organizing the coffeebreaks. We also thank Anne Reinarz, Duong Pham and Claudio Bierig for helping uswith some practical issues before and during the Workshop.
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Detailed program
Monday, February 6
9:00–9:55 Ernst P. Stephanhp-adaptive DG-FEM for Parabolic Obstacle Problems
10:00–10:55 Claudio CanutoAdaptive Fourier-Galerkin Methods
11:00–11:30 Coffee break
11:30–12:25 Alfio QuarteroniDiscontinuous approximation of elastodynamics equations
12:30–14:30 Lunch break
14:30–15:25 Monique DaugeWeighted analytic regularity in polyhedra
15:30–16:25 Jens Markus MelenkStability and convergence of Galerkin discretizations of the Helmholtz equation
16:30–17:00 Coffee break
17:00–17:55 Wolfgang HackbuschSelfadaptive Compression in Tensor Calculus
Tuesday, February 7
9:00–9:55 Ian H. SloanPDE with random coefficients as a problem in high dimensional integration
10:00–10:55 Fabio NobileStochastic Polynomial approximation of PDEs with random coefficients
11:00–11:30 Coffee break
11:30–12:00 Claude J. GittelsonPotential and Limitations of Sparse Tensor Product Stochastic Galerkin Methods
12:00–12:30 Alex BespalovA priori error analysis of stochastic Galerkin mixed finite element methods
12:30–14:30 Lunch break
14:30–15:25 Dongbin XiuA Flexible Stochastic Collocation Algorithm on Arbitrary Nodes via Interpolation
15:30–16:00 Alexander LitvinenkoComputation of maximum norm and level sets in the canonical tensor format withapplication to SPDE
16:00–16:30 Matthias Maischakp-FEM and p-BEM – achieving high accuracy
16:30–17:00 Coffee break
17:00–17:55 Ernst RankTo mesh or not to mesh: High order numerical simulation for complex structures
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Wednesday, February 8
9:00–9:55 Mark AinsworthBernstein-Bezier finite elements of arbitrary order and optimal assembly procedures
10:00–10:55 Spencer SherwinFrom h to p efficiently: balancing high or low order approximations with accuracy
11:00–11:30 Coffee break
11:30–12:25 Joachim SchoberlDomain Decomposition Preconditioning for High Order Hybrid Discontinuous GalerkinMethods 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 DemkowiczDPG method with optimal test functions. A progress report.
10:00–10:55 Paul HoustonApplication of hp-Adaptive Discontinuous Galerkin Methods to Bifurcation Phenomenain Pipe Flows
11:00–11:30 Coffee break
11:30–12:25 Dominik SchotzauExponential convergence of hp-version discontinuous Galerkin methods for elliptic prob-lems in polyhedral domains
12:30–14:30 Lunch break
14:30–15:25 Thomas P. WihlerA Posteriori Error Analysis for Linear Parabolic PDE based on hp-discontinuous GalerkinTime-Stepping
15:30–16:25 Jan S. HesthavenCertified Reduced Basis Methods for Integral Equations with Applications to Electro-magnetics
16:30–17:00 Coffee break
17:00–17:55 Ralf HiptmairConvergence of hp-BEM for the electric field integral equation
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Friday, February 10
9:00–9:55 Annalisa BuffaOn local refinement in isogeometric analysis
10:00–10:30 Lourenco Beirao da VeigaDomain decomposition methods in Isogeometric analysis
10:30–11:00 Alessandro RealiIsogeometric Collocation Methods for Elasticity
11:00–11:30 Coffee break
11:30–12:25 Giancarlo SangalliIsogeometric discretizations of the Stokes problem
12:30–14:30 Lunch break
14:30–15:00 Alexander DusterNumerical homogenization procedures in solid mechanics using the finite cell method
15:00–15:30 Zohar YosibashSimulating the mechanical response of arteries by p-FEMs
15:30–16:00 Coffee break
16:00–16:55 Yvon MadayA generalized empirical interpolation method based on moments and application
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Monday, 9:00–9:55
hp-adaptive DG-FEM for Parabolic
Obstacle Problems
Ernst P. Stephan
Insitut fur Angewandte Mathematik (IfAM), Leibniz Universitat Hannover (LUH),Germany
Joint work with:Lothar Banz (IfAM, LUH, Germany)
Parabolic obstacle problems find applications in the financial markets for pricing Amer-ican put options. We present a mixed method and an equivalent variational inequalitymethod, both based on an hp-interior penalty DG (IPDG) method combined withan hp-time DG (TDG) method, to solve parabolic obstacle problems approximatively[1]. The contact conditions are resolved by a biorthogonal Lagrange multiplier andare component-wise decoupled. These decoupled contact conditions are equivalent tofinding the root of a non-linear complementary function. This non-linear problem canin turn be solved efficiently by a semi-smooth Newton method. For the hp-adaptivitya p-hierarchical error estimator in conjunction with a local analyticity estimate is em-ployed, where the latter decides weather an h or p-refinment should be carried out. Forthe stationary problem, this leads to exponential convergence, and for the instationaryproblem to greatly improved convergence rates.Numerical experiments are given demonstrating the strengths and limitations of theapproaches.
References
[1] L. Banz hp-Finite Element and Boundary Element Methods for Elliptic, EllipticStochastic, Parabolic and Hyperbolic Obstacle and Contact Probelems, PhD-Thesis(submitted).
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Monday, 10:00–10:55
Adaptive Fourier-Galerkin Methods
Claudio Canuto
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 ellipticself-adjoint problems relies on the dynamic interplay between two fundamental stages:the refinement of the geometric decomposition into elements and the enrichment of thelocal basis within an element. While the former stage is by now well understood interms of practical realization and optimality properties, less theoretical attention hasbeen devoted to the latter one.
With the aim of shedding light on this topic, we focus on what happens in a single ele-ment of the decomposition. In order to reduce at a minimum the technical burdens, weactually 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 considera fully adaptive Fourier method, in which at each iteration of the adaptive algorithmthe Galerkin solution is spanned by a dynamically selected, finite subset of the wholeset of Fourier basis functions. The active set is determined by looking at a fixed fractionof largest (scaled) Fourier coefficients of the residual, according to the bulk-chasing (orDorfler marking) philosophy. The algorithm is proven to be convergent; in addition,exploiting the concentration properties of the inverse of the elliptic operator repre-sented in the Fourier basis, one can show that the error reduction factor per iterationtends to 0 as the bulk-chasing fraction tends to 1.
After convergence has been established, one is faced with the issue of optimality. Thisleads to the comparison of the adaptive Galerkin solution spanned by, say, N Fouriermodes, with the best N -term approximation of the exact solution. Consequently, weare led to introduce suitable sparsity classes of periodic H1-functions for which the bestN -term approximation error fulfils a prescribed decay as N tends to infinity. Theseclasses 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 algebraicrate (this occurs if the solution belongs to a certain “oblique” scale of Besov spaces ofperiodic functions, corresponding to a finite regularity), then the arguments developedby Cohen, Dahmen and DeVore and by Stevenson et al in the framework of waveletbases apply to our situation as well, and one can establish the optimality of the approx-imation without coarsening. The crucial ingredients in the analysis are the minimality
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property of the active set of degrees of freedom determined by bulk-chasing, and ageometric-series argument (essentially, the estimated number of degrees of freedomadded at each iteration is comparable to the total number of degrees of freedom addedin all previous iterations).
On the other hand, the case of a solution having (local) infinite-order or analyticalregularity is quite significant in dealing with spectral (spectral-element) methods. Forthe Navier-Stokes equations, regularity results in Gevrey classes have been first estab-lished by Foias and Temam; in these classes, the best N -term approximation error ofa function decays precisely at a sub-exponential or exponential rate. Then, the anal-ysis of an adaptive strategy becomes much more delicate, as the previous argumentsfail to apply. For instance, even for a linear operator with analytic ceofficients, it isshown that the Gevrey sparsity class of the residual is different from (actually, worsethan) the one of the exact solution. Therefore, we devise and analyse a new versionof the adptive algorithm considered before: since the choice of the next set of activedegrees of freedom is made on the basis of the sparsity of the residual, we incorporatea 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.
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Monday, 11:30–12:25
Discontinuous approximation
of elastodynamics equations
Alfio Quarteroni
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 substra-tum, from information provided by elastic wave propagations, has increased the interesttowards computational seismology. Recent developments have been focused on spectralelement methods. The reason relies on their flexibility in handling complex geometries,retaining the spatial exponential convergence for locally smooth solutions and a naturalhigh level of parallelism. In this talk, we consider a Discontinuous Galerkin spectralelement method (DGSEM) as well as discontinuous mortar methods (DMORTAR) tosimulate seismic wave propagations in three dimensional heterogeneous media. Themain advantage with respect to conforming discretizations as those based on SpectralElement Method is that DG and Mortar discretizations can accommodate discontinu-ities, 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 Ωi. Weallow this substructure decomposition to be geometrically non-conforming. Inside eachsubstructure Ωi, a conforming high order finite element space associated to a partitionThi
(Ωi) is introduced. We allow the use of different polynomial approximation degreeswithin different substructures. Applications to simulate benchmark problems as wellas realistic seismic wave propagation processes are presented.
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Monday, 14:30–15:25
Weighted analytic regularity in polyhedra
Monique Dauge
IRMAR, CNRS and Universite de Rennes 1, FRANCE
Joint work with:Martin Costabel (IRMAR, Universite de Rennes 1, FRANCE),Serge Nicaise (LAMAV, Universite de Valenciennes, FRANCE)
Elliptic boundary value problems with analytic data (coefficients, domain and righthand sides) are analytic up to the boundary [5]. Such a regularity allows exponentialconvergence of the p-version of the finite element method. This result does not hold inthe same form in domains with edges and corners.
In ca. 1988 Babuska 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 weightedanalytic classes (which they call countably normed spaces),
3. The solutions to standard elliptic boundary problems belong to such analyticclasses.
They proved results covering these three points for bi-dimensional problems (Laplace,Lame). They started the investigation of three-dimensional problems in [2, 3, 4]. Butthree-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 blowingup of the number of degrees of freedom. Exponential convergence with such dis-cretization will be obtain only if solutions belong to anisotropic analytic weightedspaces.
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 coercivevariational problems, homogeneous with constant coefficients. This completes for thefirst time point 3. of the Babuska-Guo program in polyhedra. In this talk, I present theresults of this paper. The case of Laplace-Dirichlet and Laplace-Neumann problems willserve as threads along our way in the various possible combinations of weights in two-and three-dimensional domains, and the various corresponding statements. Generalstatements will be provided as well.
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References
[1] M. Costabel, M. Dauge, and S. Nicaise, Analytic Regularity for Linear EllipticSystems 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 valueproblems 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. Babuska. Regularity of the solutions for elliptic problems on nonsmoothdomains in R3. I. Countably normed spaces on polyhedral domains. Proc. Roy.Soc. Edinburgh Sect. A, 127 (1997), No. 1, 77–126.
[4] B. Guo, I. Babuska. Regularity of the solutions for elliptic problems on nonsmoothdomains in R3. II. Regularity in neighbourhoods of edges. Proc. Roy. Soc. Edin-burgh Sect. A, 127 (1997), No. 3, 517–545.
[5] C. B. Morrey, Jr., L. Nirenberg. On the analyticity of the solutions of linear ellipticsystems of partial differential equations. Comm. Pure Appl. Math. 10 (1957), 271–290.
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Monday, 15:30–16:25
Stability and convergence of Galerkin
discretizations of the Helmholtz equation
J.M. Melenk
Vienna University of Technology
Joint work with:Stefan Sauter (University of Zurich),Maike Lohndorf (Kapsch TrafficCom)
We consider boundary value problems for the Helmholtz equation at large wave num-bers k. In order to understand how the wave number k affects the convergence proper-ties of discretizations of such problems, we develop a regularity theory for the Helmholtzequation that is explicit in k. At the heart of our analysis is the decomposition of so-lutions into two components: the first component is an analytic, but highly oscillatoryfunction and the second one has finite regularity but features wavenumber-independentbounds.
This new understanding of the solution structure opens the door to the analysis ofdiscretizations of the Helmholtz equation that are explicit in their dependence on thewavenumber k. As a first example, we show for a conforming high order finite elementmethod that quasi-optimality is guaranteed if (a) the approximation order p is selectedas 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 arisingin acoustic scattering. Also for this example, the same scale resolution conditions asin the high order finite element case suffice to ensure quasi-optimality of the Galekrindiscretization.
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Monday, 17:00–17:55
Selfadaptive Compression in Tensor Calculus
Wolfgang Hackbusch
Max-Planck-Institut Mathematik in den NaturwissenschaftenInselstr. 22, D-04103 Leipzig
The principle behind high-order methods is the wish to obtain high accuracy by asmall number of degrees of freedom. This leads, e.g., to the hp-method, where stillthe question arises how the step sizes and polynomial degrees are to be chosen opti-mally. A further question is whether polynomials should be replaced by another classof functions. A similar situation occurs for wavelet approaches.
The tensor approaches for functions in product domains (e.g, in Rd) start usually with aregular grid as known from old-fashioned difference methods. This seems to contradictthe modern approaches. However, the algorithms in tensor calculus are essentiallybased on compression principle. In the best case, the total data size nd (n grid pointsin each of the d directions) can be compressed to O(d · log n · log(1/ε)) in order toobtain an approximation of accuracy ε. It turns out that the obtainable compressionis at least as good as obtainable by hp-methods (or other high-order) methods. Theessential difference is that compression methods in tensor calculus find the suitabledistributions of h and p automatically and even chooses the suitable class of functionsin 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 logN)-quantics approximation of N − d tensors in high-dimensional numerical modeling. Constr. Approx. (2011). To appear
[4] Oseledets, Ivan V.: Approximation of matrices using tensor decomposition. SIAMJ. Matrix Anal. Appl. 31, 2130–2145 (2010)
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Tuesday, 9:00–9:55
PDE with random coefficients as a problem
in high dimensional integration
Ian H Sloan
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 finite-element methods to handle an elliptic PDE with a random field as a coefficient. A num-ber of groups have considered such problems (under headings such as polynomial chaos,stochastic Galerkin and stochastic collocation) by reformulating them as deterministicproblems in a high dimensional parameter space, where the dimensionality comes fromthe number of random variables needed to characterise the random field. In a recentpaper 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 multidimensionalintegral - together with finite- element approximation in the physical space. We userecent developments in the theory and practice of quasi-Monte Carlo integration rulesin weighted Hilbert spaces, through which rules with optimal properties can be con-structed once the weights are known. The novel feature of this work is that for the firsttime we are able to design weights for the weighted Hilbert space that achieve what isbelieved to be the best possible rate of convergence, under conditions on the randomfield that are exactly the same as in a recent paper by Cohen, DeVore and Schwab onbest N -term approximation for the same problem.
References
[1] A. Cohen, R. De Vore and C. Schwab Convergence rates of best N -term apporxi-mations 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 fora class of elliptic partial differential equations with random coefficients, submitted,2011.
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Tuesday, 10:00–10:55
Stochastic Polynomial approximation of PDEs
with random coefficients
Fabio Nobile
CSQI MATHICSE, Ecole Polytechnique Federale de Lausanne, Station 8, CH-1015Lausanne, 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 ofSaudi Arabia)
We consider the problem of numerically approximating statistical moments of the so-lution of a partial differential equation (PDE), whose input data (coefficients, forcingterms, boundary conditions, geometry, etc.) are uncertain and described by a finiteor countably infinite number of random variables. This situation includes the case ofinfinite dimensional random fields suitably expanded in e.g Karhunen-Loeve or Fourierexpansions.
We focus on polynomial chaos approximations of the solution with respect to theunderlying random variables and review common techniques to practically computesuch polynomial approximation by Galerkin projection, Collocation on sparse grids orregression methods from random evaluations.
We discuss in particular the proper choice of the polynomial space both for linear ellipticPDEs with random diffusion coefficient and second order hyperbolic equations withrandom piecewise constant wave speed. Numerical results showing the effectivenessand limitations of the approaches will be presented as well.
References
[1] G. Migliorati, F. Nobile, E. von Schwerin, and R. Tempone, “Analysis of the discreteL2 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 forthe second order wave equation with a discontinuous random speed,” MOX-Report36-2011, Department of Mathematics, Politecnico di Milano, Italy, 2011. submitted.
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[3] J. Back, F. Nobile, L. Tamellini, and R. Tempone, “On the optimal polynomialapproximation 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. Babuska, F. Nobile, and R. Tempone, “A stochastic collocation method for ellip-tic partial differential equations with random input data,” SIAM Review, vol. 52,pp. 317–355, June 2010.
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Tuesday, 11:30–12:00
Potential and Limitations of Sparse Tensor Product
Stochastic Galerkin Methods
Claude J. Gittelson
Purdue University
Sparse tensor product constructions are known to be more efficient than their fulltensor product counterparts in many applications. In case of low regularity, spectralapproximations for random differential equations form an exception to this rule.
We consider an elliptic PDE with a random diffusion coefficient, which we assume tobe expanded in a series. The solution to such an equation depends on the coefficientsin this series in addition to the spatial variables. Spectral discretizations typically usetensorized polynomials to represent the parameter-dependence of the solution. Eachcoefficient of the solution with respect to such a polynomial basis is a function of thespatial variable, and can be approximated by finite elements.
Since the importance of the polynomial basis functions can vary greatly, it is naturalto expect a gain in efficiency if the coefficients are approximated in different finite ele-ment spaces. We show under what conditions such a multilevel construction reaches ahigher convergence rate than approximations employing just a single spatial discretiza-tion. Sparse tensor products, which impose additional structure on the multilevelapproximation, can always essentially attain the optimal convergence rate.
References
[1] C.J. Gittelson, Convergence rates of multilevel and sparse tensor approximationsfor a random elliptic PDE, Submitted.
21
Tuesday, 12:00–12:30
A priori error analysis of stochastic Galerkin
mixed finite element methods
Alex Bespalov
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 prob-lems with random data (i.e., coefficients, sources, boundary conditions, geometry etc.).These methods combine conventional Galerkin finite elements on the (spatial) compu-tational domain with spectral approximations in the stochastic variables. Whilst thereexists a large body of research work on numerical approximation of primal formulationsfor elliptic PDEs with random data, the case of mixed variational problems is not sowell developed.
In this talk we will address the issues involved in the error analysis of stochasticGalerkin approximations to the solution of a first order system of PDEs with randomcoefficients. First, we approximate the random input by using spectral expansions in Mrandom variables and transform the variational saddle point problem to a parametricdeterministic one. Approximations to the latter problem are then constructed by com-bining mixed hp finite elements on the computational domain with M -variate tensorproduct polynomials of order k = (k1, k2, . . . , kM). We will discuss the inf-sup stabilityand well-posedness of the continuous and finite-dimensional problems, the regularityof solutions with respect to the M parameters describing the random coefficients, anda priori error bounds for stochastic Galerkin approximations in terms of discretisationparameters 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
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 uncertaintyquantification and stochastic computing of large and complex systems. Though theidea has been explored in the past, its popularity is largely due to the recent advanceof employing high-order nodes such as sparse grids. These nodes allow one to conductUQ 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 forstochastic collocation method can grow fast and render the method computationallyprohibitive. Such kind of growth is especially severe when the nodal construction isstructured, e.g., tensor grids, sparse grids, etc. One way to alleviate the difficulty is toemploy 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 workwith arbitrary number of nodes on arbitrary locations. Another strong motivation isthe practical restriction one may face. In many cases one can not conduct simulationsat the desired nodes.
In this work we present an algorithm that allows one to construct high-order polyno-mial responses based on stochastic collocation on arbitrary nodes. The method is basedon constructing a “correct” polynomial space so that multi-dimensional polynomial in-terpolation can be constructed for any data. We present its rigorous mathematicalframework, its practical implementation details, and its applications in high dimen-sions.
23
Tuesday, 15:30–16:00
Computation of maximum norm and level sets
in the canonical tensor format with application
to SPDE
Alexander Litvinenko
Technische Universitat 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 Universitat 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 fieldsvia KLE and PCE, the obtained high dimensional operator can be approximated viasums of elementary tensors. This tensors representation can be effectively used forcomputing different values of interest, such as maximum norm, level sets and cumula-tive distribution function. The basic concept of the data analysis in high dimensions isdiscussed on tensors represented in the canonical format, however the approach can beeasily used in other tensor formats. As an intermediate step we describe efficient itera-tive algorithms for computing the characteristic and sign functions as well as pointwiseinverse in the canonical tensor format. Since during majority of algebraic operationsas well as during iteration steps the representation rank grows up, we use lower-rankapproximation and inexact recursive iteration schemes.
References
[1] M. Espig, W. Hackbusch, A. Litvinenko, H. G. Matthies and E. Zander, EfficientAnalysis of High Dimensional Data in Tensor Formats, Technische UniversitatBraunschweig, 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
Brunel University, Uxbridge UB8 3PH, UK
It is well known, that we can achieve exponential fast convergence by approximatingthe 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 surprisinglydifficult, due to requirements on memory, solution time and the necessary numericalprecision.
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-precisionpackage [1] and discuss implementation issues which arise from considerable more timeand memory consuming arithmetic operations.
We will compare standard double precision (hardware supported), quad precision (com-piler supported) and higher precision (additional Software package [1]). We will presentexamples achieving errors with 1e-30 or less using p- and hp fem-bem methods in 1,2and 3 dimensions.
References
[1] David H. Bailey, Yozo Hida, Xiaoye S. Li, and Brandon Thompson, ARPREC: AnArbitrary Precision Computation Package, Technical Report LBNL-53651, 2002
[2] B. Guo, and I. Babuska, 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 hp-version for the boundary element method on open surfaces, Numerische Mathe-matik, 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
Chair for Computation in Engineering, Technische Universitat Munchen, Arcisstrasse21, 80333 Munchen, Germany
Joint work with:Stefan Kollmannsberger, Dominik Schillinger, Christian Sorger (Chair for
Computation in Engineering, Technische Universitat Munchen, Arcisstrasse 21, 80333Munchen, Germany)
The paper will give an overview over some recent work on high order finite elementmeshing for thin-walled structures as well as on a high order variant of fictitious do-main 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 mod-els and is not depending on dimensional reduction like in shell or plate models. Totake full advantage of this method it is yet necessary to develop special mesh gener-ating techniques which discretize the exact 3-dimensional curved geometry and not afacetted mid-surface of the structure. In the second part of the presentation the FiniteCell Method (FCM), an extension of high order finite element methods to a fictitiousdomain approach will be discussed. It completely avoids meshing, yet converges ex-ponentially in energy norm for smooth problems. This approach turns out to be veryefficient and accurate and can even be used for a real-time simulation of complex solidstructures. Very recent results on an extension of the Finite Cell Method by an adap-tive, hierarchic refinement allows a tight connection between geometric modeling andnumerical analysis for thin-walled as well as for massive structures.
26
Wednesday, 9:00–9:55
Bernstein-Bezier finite elements of arbitrary order
and optimal assembly procedures
Mark Ainsworth
University of Strathclyde
Joint work with:G. Andriamaro and O. Davydov
Algorithms are presented that enable the element matrices for the standard finite el-ement space, consisting of continuous piecewise polynomials of degree n on simplicialelements in Rd, to be computed in optimal complexity O(n2d). The algorithms (i) takeaccount 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 func-tions at quadrature points (although these can be used if desired). The elements arebased on Bernstein-Bezier polynomials and are the first to achieve optimal complexityfor 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
Department of Aeronautics, Imperial College London, [email protected]
When approximating a PDE problem a reasonable question to ask is what is the idealdiscretisation order for a desired computational error? Although we typically un-derstand the answer in rather general terms there are comparatively little quantitiveresults? The issues behind this question are partly captured in the figure above wherewe observe the wall shear stress pattern around two small branching vessels in andanatomically realistic mode of part of the descending aorta. In the left image we ob-serve a linear finite element approximation on a fine mesh and in the right image acoarse mesh approximation using a 6th order polynomial expansion is applied. Thehigh order discretisation is much smoother but this accuracy typically is associatedwith a higher computational cost. Although the low order approximation looks morepixelated it does however capture all the salient features of the wall shear stress pat-terns.
The spectral element and hp finite element methods methods can be considered asbridging the gap between the – traditionally low order – finite element method on oneside and spectral methods on the other side. Consequently, a major challenge whicharises in implementing the spectral/hp element methods is to design algorithms thatperform efficiently for both low- and high-order discretisations, as well as discretisationsin the intermediate regime.
In this presentation, we explain how the judicious use of different implementationstrategies can be employed to achieve high efficiency across a range of polynomial
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orders [1, 2]. Further, based upon this efficient implementation, we analyse whichspectral/hp discretisation minimises the computational cost to solve elliptic steadyand unsteady model problems up to a predefined level of accuracy. Finally we discusshow the lessons learned from this study can be encapulated into a more generic libraryimplementation.
References
[1] P.E.J. Vos, S.J. Sherwin and R.M. Kirby, From h to p efficiently: Implementingfinite and spectral/hp element discretisations to achieve optimal performance atlow 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. Kirbyand P.H.J. Kelly, From h to p efficiently: Strategy selection for operator evaluationon 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 Schoberl
Vienna University of Technology
Hybrid discontinuos Galerkin methods are popular discretization methods in applica-tions from fluid dynamics and many others. Often large scale linear systems arisingfrom elliptic operators have to be solved. We show that standard p-version domaindecomposition techniques can be applied, but we have to develop new technical toolsto prove poly-logarithmic condition number estimates, in particular on tetrahedralmeshes.
30
Thursday, 9:00–9:55
DPG method with optimal test functions.
A progress report.
Leszek Demkowicz
University of Texas at Austin
I will present a progress report on the Discontinuous Petrov-Galerkin method proposedby 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 elementlevel. If the error in approximating the optimal test functions is negligible, the methodAUTOMATICALLY guarantees the discrete stability, provided the continuous problemis well posed. And this holds for ANY linear problem. The result is shocking until onerealizes that we are working with an unconventional least squares method. The twistlies in the fact that the residual lives in a dual space and it is computed using dualnorms.
The method turns out to be especially suited for singular perturbation problems whereone strives not only for stability but also for ROBUSTNESS, i.e. a stability UNIFORMwith respect to the perturbation parameter. Critical to the construction of a robustversion of the DPG method is the choice of the test norm that should imply the uniformstability.
I will report on recent results for two important model problems: convection-dominateddiffusion and linear acoustics equations [3, 4, 5]. I will attempt to outline a generalstrategy for selecting the optimal test norm and will focus on the task of approximatingthe optimal test functions and effects of such an approximation [6].
References
[1] L. Demkowicz and J. Gopalakrishnan. A Class of Discontinuous Petrov-GalerkinMethods. 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 ExplicitAnalysis for a DPG Method for the Multidimensional Helmholtz Equation”, ICESReport 2011-24, CMAME, in print.
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[4] L. Demkowicz, N. Heuer, Robust DPG Method for Convection-Dominated DiffusionProblems. ICES Report 2011-33, submitted to SIAM J. Num. Anal.
[5] L. Demkowicz, J. Gopalakrishnan, I. and Muga, D. Pardo, A Pollution Free DPGMethod for Multidimensional Helmholtz Equation, in preparation.
[6] J. Gopalakrishnan and W. Qiu. An Analysis of the Practical DPG Method. IMAReport 2011-2374.
32
Thursday, 10:00–10:55
Application of hp–Adaptive Discontinuous Galerkin
Methods to Bifurcation Phenomena in Pipe Flows
Paul Houston
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 re-finement of discontinuous Galerkin finite element approximations of the bifurcationproblem associated with the steady incompressible Navier–Stokes equations. Particu-lar attention is given to the reliable error estimation of the critical Reynolds numberat which a steady pitchfork bifurcation occurs when the underlying physical systempossesses rotational and reflectional or O(2) symmetry. Here, computable a posteri-ori error bounds are derived based on employing the generalization of the standardDual Weighted Residual approach, originally developed for the estimation of targetfunctionals of the solution, to bifurcation problems; see [1, 2, 3] for details. Numericalexperiments highlighting the practical performance of the proposed a posteriori errorindicator on h– and hp–adaptively refined computational meshes are presented. Here,particular attention is devoted to the problem of flow through a cylindrical pipe witha 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 andA Posteriori Error Control for Bifurcation Problems I: The Bratu Problem. Com-munications 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 APosteriori Error Control for Bifurcation Problems II: Incompressible fluid flow inOpen Systems with Z2 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 andA Posteriori Error Control for Bifurcation Problems III: Incompressible fluid flowin 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 Schotzau
Mathematics Department, University of British Columbia, Vancouver, BC V6T 1Z2,Canada
Joint work with:Christoph Schwab (Seminar of Applied Mathematics, ETH Zurich, 8092 Zurich,
Switzerland, email: [email protected]),Thomas Wihler (Mathematisches Institut, Universitat Bern, 3013 Bern, Switzerland,
email: [email protected])
We design and analyze hp-version interior penalty (IP) discontinuous Galerkin (dG)finite element methods for the numerical approximation of linear second order ellipticboundary-value problems in three dimensional polyhedral domains. The methods arebased on using hexahedral meshes that are geometrically and anisotropically refinedtowards corners and edges. Similarly, the local polynomial degrees are increased linearlyand possibly anisotropically away from singularities. We prove that the methods arewell-defined for problems with singular solutions and are stable under the proposedhp-refinements. Finally, we show that the hp-dG methods lead to exponential rates ofconvergence in the number of degrees of freedom for problems with piecewise analyticdata.
References
[1] D. Schotzau, C. Schwab, and T. Wihler: hp-dGFEM for second order elliptic prob-lems in polyhedra I: Stability and quasioptimality on geometric meshes, submitted.
[2] D. Schotzau, C. Schwab, and T. Wihler: hp-dGFEM for second order ellipticproblems 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
Mathematics Institute, University of Bern, Switzerland
Joint work with:Omar Lakkis (U of Sussex, UK),
Manolis Georgoulis (U of Leicester, UK)Dominik Schotzau (UBC), Canada
We consider full discretizations in time and space of linear parabolic PDE. More pre-cisely, in order to discretize the time variable the hp-discontinuous Galerkin time step-ping scheme from [3] is employed. Furthermore, conforming variational methods areapplied to approximate the problem in space.
The focus of the talk is the development of an L2(H1)-based a posteriori error analysis(in time and space) for the given class of schemes. Here, the key point is the combina-tion of an hp-time reconstruction (see [4], cf. also [2]) and of an elliptic reconstructionin space (see [1]). The resulting a posteriori error estimate is expressed in terms of thefully discrete solution.
References
[1] C. Makridakis and R. H. Nochetto, Elliptic reconstruction and a posteriori errorestimates 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 orderdissipative methods for parabolic problems, Numerische Mathematik, 104 (2006),489–514.
[3] D. Schotzau and C. Schwab, Time discretization of parabolic problems by the hp-version of the discontinuous Galerkin finite element method, SIAM Journal onNumerical Analysis, 38 (2000), 837–875.
[4] D. Schotzau and T. P. Wihler, A posteriori error estimation for hp-version time-stepping methods for parabolic partial differential equations, Numerische Mathe-matik, 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
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 isused extensively throughout science and engineering to enable the fast/real-time/subscalemodeling of complex systems for control, design, prediction purposes of multi-scaleproblems, uncertainty quantification etc. Such models, while often successful and ofundisputed value, are, however, often heuristic in nature and the validity and accuracyof 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. Theseare used widely to model a variety of important problems in acoustics, heat conduction,and electromagnetics and the analysis and computational techniques have been welldeveloped during the last decades. However, for many-query situations the directsolution of these models may be prohibitive and the development of reduced modelsof significant value. In the context of integral equations for electromagnetic scattering,variations over frequency, sources, observations etc, can be substantial, yet a rapidevaluation has important application in design and detection applications.
We discuss the development of certified reduced basis methods for integral equations toenable a rapid and error-controlled evaluation of the scattering over parametric varia-tion. During this talk we outline the basic elements of a certified reduced basis methodand highlight specific issues related to integral equations caused by their non-affinenature and the need to attend to computational complexity due to large parametervariations.
We continue by discussing how these reduced models can be combined iteratively tosolve larger multi-object problems without directly assembling the full physical prob-lem. Time permitting we extend this discussion to the high-dimensional parametrizedproblem and discuss ways of achieving accurate parametric compression in the reducedmodel using ANOVA expansions.
36
Thursday, 17:00–17:55
Convergence of hp-BEM for the electric field
integral equation
Ralf Hiptmair
ETH Zuich
Joint work with:A. Bespalov (University of Manchester),
N. Heuer (Pontificia Universidad Catolica 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 Galerkindiscretization based on divΓ -conforming Raviart-Thomas boundary elements (BEM)of locally variable polynomial degree on shape-regular surface meshes. We establishasymptotic quasi-optimality of Galerkin solutions on sufficiently fine meshes or forsufficiently high polynomial degree.
References
[1] A. Bespalov, N. Heuer, and R. Hiptmair, Convergence of natural hp-BEM for theelectric 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
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 connectionbetween numerical simulation of physical phenomena and the Computer Aided Designsystems. Indeed, the ultimate goal is to eliminate or drastically reduce the approx-imation 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 usingB-Splines or Non Uniform Rational B-Splines for the geometry description as well asfor the representation of the unknown fields. The use of Spline or NURBS functions,together with isoparametric concepts, results in an extremely successfully idea andpaves the way to many new numerical schemes enjoying features that would be ex-tremely hard to achieve within a standard finite element framework. One of the bigchallenge in the context of isogeometric analysis is the possibility to break the tensorproduct structure of Splines and so design adaptive methods. T-splines, introducedby T. Sederberg et al. in 2004, have been used to this aim in geometric modeling anddesign, 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 ourexperience 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 areDual-Compatible. Tech. Rep. IMATI CNR, 2012.
[2] A. Buffa, D. Cho, M. Kumar, Characterization of T-splines with reduced continuityorder on T-meshes, Comput. Methods Appl. Mech. Engrg., Volumes 201204, 1January 2012, pp. 112-126.
[3] L Beirao da Veiga, A. Buffa, D. Cho, G. Sangalli, Isogeometric analysis usingT-splines on two-patch geometries, Computer Methods in Applied Mechanics andEngineering, Vol. 200 (21-22), pp.1787-1803, (2011)
38
Friday, 10:00–10:30
Domain decomposition methods
in Isogeometric analysis
Lourenco Beirao da Veiga
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 nu-merical methodology for the analysis of PDE problems, that adopts NURBS basisfunctions for the description of the discrete space and allows for an exact descriptionof CAD-type geometries.
The discrete systems produced by isogeometric methods are better conditioned thanthe systems produced by standard finite elements or finite differences, but their condi-tioning can still degenerates rapidly for decreasing mesh size h, increasing polynomialdegrees p and increasing coefficient jumps in the elliptic operator. Therefore the de-sign and analysis of efficient iterative solvers for isogeometric analysis is a challengingresearch topic, particularly for three-dimensional problems with discontinuous coeffi-cients. The global high regularity of the adopted NURBS functions introduces a newset 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 IsogeometricAnalysis concentrating our efforts on a model elliptic problem. We will consider bothan Overlapping Schwarz [1] method and a BDDC (i.e. non overlapping) method [2]. Inboth cases we will show theoretical condition number bounds in terms of the mesh sizesh,H (such as the scalability property) and numerical tests that confirm and extend interms of the polynomial order p and regularity index k the theoretical predictions.
The theoretical and numerical results of the present talk are strongly encouraging andshow that domain decomposition seems to be a very good choice for preconditioningand solving problems in Isogeometric Analysis. Clearly more work needs to be accom-plished, both in terms of extension to more complex problems and of development (andcomparison with) different preconditioning and solving schemes.
39
References
[1] L. Beirao da Veiga, D. Cho, L. F. Pavarino, S. Scacchi, Overlapping SchwarzMethods for Isogeometric Analysis, to appear on Siam J. Numer. Anal. and PreprintIMATI-CNR (2011).
[2] L. Beirao da Veiga, D. Cho, L. F. Pavarino, S. Scacchi, BDDC preconditioners forIsogeometric Analysis, submitted for publication and Preprint IMATI-CNR (2012).
[3] J. A. Cottrell, T. J. R. Hughes, Y. Basilevs, Isogeometric Analysis: Toward Inte-gration of CAD and FEA, Wiley, Chichester, U.K., 2009
[4] A. Toselli, O. B. Widlund, Domain Decomposition Methods: Algorithms and The-ory, Computational Mathematics, Vol. 34. Springer-Verlag, Berlin, 2004.
40
Friday, 10:30–11:00
Isogeometric Collocation Methods for Elasticity
Alessandro Reali
Structural Mechanics Department, University of Pavia
Joint work with:Ferdinando Auricchio (University of Pavia),
Lourenco Beirao 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] tobridge the gap between Computational Mechanics and Computer Aided Design (CAD).The key feature of IGA is to extend the finite element method representing geometry byfunctions, such as Non-Uniform Rational B-Splines (NURBS), which are used by CADsystems, and then invoking the isoparametric concept to define field variables. Thus,the computational domain exactly reproduces the NURBS description of the physicaldomain. Moreover, numerical testing in different situations has shown that IGA holdsgreat promises, with a substantial increase in the accuracy-to-computational-effort ratiowith respect to standard finite elements, also thanks to the high regularity propertiesof the employed functions.
In the framework of NURBS-based IGA, collocation methods have been recently pro-posed by Auricchio et al. [3], constituting an interesting high-order low-cost alternativeto standard Galerkin approaches. Such techniques have also been successfully appliedto elastostatics and explicit elasodynamics [4].
In this work, after an introduction to isogeometric collocation methods, we move to thesolution of elasticity problems and present in detail the results discussed in [4]. Partic-ular attention is devoted to the imposition of boundary conditions (of both Dirichletand Neumann type) and to the treatment of the multi-patch case. Also the develop-ment of explicit high-order (in space) collocation methods for elastodynamics is hereconsidered and studied. Several numerical experiments are presented in order to showthe good behavior of these approximation techniques.
References
[1] Hughes, T.J.R., Cottrell, J.A., Bazilevs, Y., Isogeometric analysis: CAD, finiteelements, NURBS, exact geometry, and mesh refinement, Computer Methods inApplied Mechanics and Engineering, 194, 4135-4195 (2005).
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[2] Cottrell, J.A., Hughes, T.J.R., Bazilevs, Y., Isogeometric Analysis. Towards inte-gration of CAD and FEA. Wiley, (2009).
[3] Auricchio, F., Beiro da Veiga, L., Hughes, T.J.R., Reali, A., Sangalli, G., Isoge-ometric Collocation Methods, Mathematical Models and Methods in Applied Sci-ences, 20, 2075-2107 (2010).
[4] Auricchio, F., Beiro da Veiga, L., Hughes, T.J.R., Reali, A., Sangalli, G., Isogeo-metric collocation for elastostatics and explicit dynamics, ICES Report (2012).
42
Friday, 11:30–12:25
Isogeometric discretizations of the Stokes problem
Giancarlo Sangalli
Dipartimento di Matematica, Universita di Paviaand IMATI-CNR “Enrico Magenes”, Pavia
Joint work with:Andrea Bressan (Dipartimento di Matematica, Universita di Pavia),
Annalisa Buffa (IMATI-CNR “Enrico Magenes”, Pavia),Carlo de Falco (MOX, Milano),
Rafael Vazquez (IMATI-CNR “Enrico Magenes”, Pavia)
Isogeometric analysis has been introduced in 2005 by T.J.R. Hughes and co-authorsas a novel technique for the discretization of Partial Differential Equations (PDE).This technique is having a growing impact on several fields, from fluid dynamics, tostructural 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 ofthe unknown fields. Splines and NURBS offer a flexible set of basis functions forwhich mesh refinement and degree elevation are very efficient. Beside the fact thatin isogeometric analysis one can directly treat geometries described by Splines andNURBS parametrizations, these functions are interesting in themselves since they easilyallow global smoothness beyond the classical C0-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, well-known in the finite element context, to the isogeometric context. These isogeometricTaylor-Hood elements are based on p+ 1 degree NURBS velocity approximation and pdegree NURBS pressure approximation. The novelty, in the isogeometric framework, isthat Cr 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 thatprovides an exact divergence-free discrete solution. This is possible thanks to thecommuting de Rham digram property that this element fulfils (see [5, 4]).
References
[1] Andrea Bressan. Isogeometric regular discretization for the Stokes problem. IMAJournal of Numerical Analysis, 2010.
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[2] Andrea Bressan, Giancarlo Sangalli Isogeometric discretizations of the Stokes prob-lem: stability analysis by the macroelement technique. IMA Journal of NumericalAnalysis, submitted.
[3] A. Buffa, C. de Falco, and G. Sangalli. Isogeometric analysis: stable elements forthe 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. Vazquez. Isogeometric discrete differentialforms in three dimensions. SIAM Journal on Numerical Analysis, 49:818, 2011.
[5] A. Buffa, G. Sangalli, and R. Vazquez. Isogeometric analysis in electromagnetics:B-splines approximation. Computer Methods in Applied Mechanics and Engineer-ing, 199(17-20):1143–1152, 2010.
[6] J.A. Cottrell, T.J.R. Hughes, and Y. Bazilevs. Isogeometric analysis: toward inte-gration of CAD and FEA. John Wiley & Sons Inc, 2009.
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Friday, 14:30–15:00
Numerical homogenization procedures in solid
mechanics using the finite cell method
Alexander Duster
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 Munchen)
A numerical procedure for the homogenization of heterogeneous and foamed materi-als is presented. Based on this approach, effective material properties are determinedwhich can be utilized in finite element computations of engineering structures. Thenumerical homogenization strategy is applied to model and compute three-dimensionalcomposites and sandwich structures which are composed of a foamed core covered bytwo faceplates. The homogenization is performed by applying the Finite Cell Method(FCM) which can be interpreted as a combination of a fictitious domain method withhigh-order finite elements. Applying the FCM dramatically simplifies the meshing pro-cess, since a simple Cartesian grid is used and the geometry of the microstructure andthe different material properties are taken care of during the integration of the stiffnessmatrices of the cells. In this way, the effort is shifted from mesh generation towards thenumerical integration, which can be performed adaptively in a fully automatic way. Itwill be demonstrated by several numerical examples that this approach can be appliedto analyze a computer-aided design of a new heterogeneous material or data obtainedby a three-dimensional computer-tomography of the material of interest.
References
[1] A. Duster, H.-G. Sehlhorst, and E. Rank, Numerical homogenization of hetero-geneous and cellular materials utilizing the Finite Cell Method, ComputationalMechanics, (2012), DOI: 10.1007/s00466-012-0681-2.
[2] A. Duster, J. Parvizian, Z. Yang, and E. Rank, The Finite Cell Method for 3Dproblems of solid mechanics, Computer Methods in Applied Mechanics and Engi-neering, (2008), 197:3768–3782.
[3] J. Parvizian, A. Duster, and E. Rank, Finite Cell Method: h- and p-extension forembedded domain problems in Solid Mechanics, Computational Mechanics, (2007),41:121-133.
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Friday, 15:00–15:30
Simulating the mechanical response of arteries
by p-FEMs
Zohar Yosibash
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 re-sponse is of major interest and attracted a significant amount of research, mainly relatedto its passive response. Herein we present a high-order FE method for the treatmentof the highly non-linear, almost-incopressible hyper-elastic constitutive model for thecoupled passive-active response based on a transversely isotropic strain-energy-density-function (SEDF).
More specifically, we first address SEDF of the passive response [1]. The new p-FE al-gorithms developed to expedite the numerical computations are presented. Numericalexamples are provided to demonstrate the efficiency of the p-FEMs compared to clas-sical 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 significantinfluence on the overall mechanical response of the artery wall, it has been scarcelyinvestigated because of the difficulty of representing such an influence in a constitutivemodel and lack of experimental evidence that can validate such models. This activeresponse is strongly coupled with the passive response because of the mutual interaction(smooth muscle cells react as a function of the mechanical response). We incorporatedthe 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 in-compressible materials under finite deformations with applications to arteries sim-ulation, International Journal for Numerical Methods in Engineering, 88, (2011),1152–1174.
[2] Zohar Yosibash and Elad Priel, Artery active mechanical response: High orderfinite element implementation and investigation, Submitted for publication, (2012)
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Friday, 16:00–16:55
A generalized empirical interpolation method
based on moments and application
Yvon Maday
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 asimple and efficient construction of an interpolation process on manifold of small Kol-mogorov width. This approach has attracted some attention in the frame of the reducedbasis method and more generally reduced order approximation for the approximationof nonlinear PDE’s. The application range is nevertheless much larger.
We propose in this talk a generalization of this concept with applications to combinedassimilation-simulation technics for complex phenomenon.
References
[1] Barrault, Maxime and Maday, Yvon and Nguyen, Ngoc Cuong and Patera, An-thony T. An ‘empirical interpolation’ method: application to efficient reduced-basisdiscretization of partial differential equations, Comptes Rendus Mathematique.Academie des Sciences. Paris, 339 (2004) 9, pp 667–672
[2] Maday, Yvon and Nguyen, Ngoc Cuong and Patera, Anthony T. and Pau, GeorgeS. H. A general multipurpose interpolation procedure: the magic points, Commun.Pure Appl. Anal., 8, (2009), 1, pp 383–404.
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List of Participants
Ainsworth Mark University of Strathclyde
Andreev Roman ETH Zurich
Banz Lothar Leibniz Universitat Hannover
Bartha Ferenc University of Bergen
Beirao da Veiga Lourenco Universita di Milano
Bespalov Alexey University of Manchester
Beuchler Sven Universitat Bonn
Bierig Claudio Universitat Bonn
Brandsmeier Holger ETH Zurich
Buffa Annalisa CNR, Pavia
Canuto Claudio Politecnico di Torino
Chernov Alexey Universitat Bonn
Costabel Martin Universite de Rennes 1
Dauge Monique Universite de Rennes 1
Demkowicz Leszek University of Texas at Austin
Duster Alexander TU Hamburg-Harburg
Egger Herbert TU Munchen
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 Zurich
Houston Paul University of Nottingham
Klose Roland CST AG, Darmstadt
Litvinenko Alexander TU Braunschweig
Maday Yvon Universite Pierre et Marie Curie
Maischak Matthias Brunel University, Uxbridge
Melenk Jens Markus TU Wien
Merabet Ismail Universite de Valenciennes
Nicaise Serge Universite de Valenciennes
Nobile Fabio EPFL Lausanne
Pham Duong Thanh Universitat Bonn
Quarteroni Alfio EPFL Lausanne
Rank Ernst TU Munchen
Reali Alessandro Universita di Pavia
Reinarz Anne Universitat Bonn
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Sangalli Giancarlo Universita di Pavia
Schmidt Kersten TU Berlin
Schoberl Joachim TU Wien
Schotzau Dominik University of British Columbia, Vancouver
Schroder Andreas HU Berlin
Schwab Christoph ETH Zurich
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 Universitat Hannover
Valli Alberto Universita di Trento
Weggler Lucy Universitat des Saarlandes
Wihler Thomas Universitat Bern
Xiu Dongbin Purdue University
Yosibash Zohar Ben-Gurion University, Beer-Sheva
Zaglmayr Sabine CST AG, Munchen
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