Austrian NumericalAnalysis Day

4-5 May 2017

Salzburg

Department of Mathematics

University of Salzburg

Austrian NumericalAnalysis Day

4-5 May 2017Salzburg

EditorsLothar BanzRaoul KutilAndreas Schroder

OrganizationLothar BanzLothar.Banz@sbg.ac.at

Andreas SchroderAndreas.Schroeder@sbg.ac.at

Department of MathematicsParis Lodron University of SalzburgHellbrunnerstr. 345020 Salzburg, Austria

LocationParis Lodron University of SalzburgDepartment of MathematicsFaculty of Natural Science(Gruner Horsaal - HS403)Hellbrunnerstr. 345020 Salzburg, Austria

Supporting Organizations

Paris Lodron University of Salzburg

Department of Mathematics

Preface

Following in the spirit and the tradition of previous events in this series of workshops, the 13thAustrian Numerical Analysis Day will be organized by and will take place at the Departmentof Mathematics of the Paris Lodron University of Salzburg on 4th and 5th of May 2017.

The goal of this workshop is to inform about research activities in the fields of numerical analysisand applied mathematics. Scientists from Austrian universities and other research institutions inparticular are invited to present their research and discuss their ideas. Apart from strengtheningalready well established contacts this annual workshop should also provide an opportunity tostart new collaborations.

Lothar Banz and Andreas Schroder

Salzburg, May 2017

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Contents

Internet Access 7

Conference Dinner 8

Program 10

Abstracts 12

Benjamin Stadlbauer: Brownian-dynamics simulations of nonopore protein sensing 12

Gregor Mitscha-Baude: Simulation of nanopores with the Poisson-Nernst-Plack-Stokes and adaptive finite elements . . . . . . . . . . . . . . . . . . . . . . . . . . 13

Stefan Rigger: Approximation of Multisymmetric Functions . . . . . . . . . . . . . . 14

Gudmund Pammer: Computing Cubature Formulas for Multisymmetric Functionsand Applications to Stochastic Partial Differential Equations . . . . . . . . . . . 15

Amirreza Khodadadian: Optimal multi-level Monte Carlo method for the stochasticdrift-diffusion-Poisson system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

Boaz Blankrot: Multiple scattering approach for dielectric metamaterial analysis . . 17

Lukas Kogler: ASC-AMG, a parallel AMG-solver for Netgen/NGSolve . . . . . . . . 18

Bernd Schwarzenbacher: Algebraic Multigrid for Maxwell’s Equations . . . . . . . 19

Daniel Jodlbauer: Robust and Efficient Solvers for Fluid-Structure Interaction . . . 20

Markus Gasteiger: ADI type preconditioners for the steady state inhomogeneousVlasov equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

Gerhard Kitzler: A tensor product framework for kinetic equations . . . . . . . . . 22

Darian M. Onchis: Numerical considerations of consistency and stability in spline-type spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

Christian Gerhards: Modeling and Numerical Aspects of Inverse Problems in Geo-magnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

Markus Schobinger: Simulation of Eddy Currents in an Iron Ring Core Using aMulti-Scale Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

Mario Luiz Previatti de Souza: Convergence result for IRGNM type method undera tangential cone condition in Banach space . . . . . . . . . . . . . . . . . . . . . 26

Paolo Di Stolfo: Dual weighted residual error estimation for the finite cell method . 27

Bernhard Endtmayer: Adaptive Mesh Refinement for Multiple Goal Functionals . 28

Joscha Gedicke: Residual-based a posteriori error analysis for symmetric mixedArnold-Winther FEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

Thomas Fuhrer: On the DPG method for Signorini problems . . . . . . . . . . . . . 30

Nina Ovcharova: Numerical methods for nonmonotone contact problems in contin-uum mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

Markus Faustmann: Local convergence of the boundary element method on poly-hedral domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

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Thomas Apel: Superconvergent graded meshes: First results . . . . . . . . . . . . . 33

Philip Lukas Lederer: Polynomial robust stability analysis for H(div)-conformingfinite elements for the Stokes equations . . . . . . . . . . . . . . . . . . . . . . . . 34

Olaf Steinbach: Regularization error estimates for distributed control problems . . . 35

Clemens Hofreither: A Black-Box Algorithm for Fast Matrix Assembly in Isogeo-metric Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

Linus Wunderlich: Isogeometric mortar methods in solid mechanics . . . . . . . . . 37

Svetlana Matculevich: Functional a posteriori error estimates and adaptivity forIgA schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

Ioannis Toulopoulos: Time Discontinuous Galerkin Multipatch Isogeometric Anal-ysis of Parabolic Diffusion Problems . . . . . . . . . . . . . . . . . . . . . . . . . 39

Stefan Dohr: Space-time boundary element methods for the heat equation . . . . . . 40

Marco Zank: Space-Time Boundary Element Method for the Wave Equation . . . . 41

Bernhard Stiftner: Linear second order implicit-explicit time-integration of the(eddy-currents-)Landau-Lifshitz-Gilbert equation . . . . . . . . . . . . . . . . . . 42

Harald Hofstatter: Time propagators for Schrodinger-type equations with expensive-to-evaluate nonlinear part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

List of participants 45

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Internet Access

During the Austrian Numerical Analysis Day there will be free internet access in the lectureroom during the seminar. Please enter the following in the login mask:

SSID: Plus EventUser: AUNUDAPassword: yalana98!

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Conference Dinner

For the conference dinner, we have reserved a table at a nearby restaurant Sternbrau that pro-vides a three course meal. The joint dinner at the traditional Austrian restaurant is an optionalopportunity to discuss various topics.

Please note that the coupon included in the workshop documents covers the dinner and thefirst drink. The menu is:

Strong beef broth with pancake stripes

Roast beef in onion gravy with spaetzle and green beansorRoasted brook trout filet on creamy kohlrabi with bulgur and braised tomato

Sweet cheese dumplings with sour cherry sauce

Dinner: Sternbrau – May 4, 2017 starting at 7:30pm

Griesgasse 23, 5020 Salzburg.http://www.sternbrau.com/en/

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How to get to Sternbrau:

It will take you approx. 6 minutes to get from the building where the conference is taking placeto the bus stop ”Faistauergasse” on the main road (Alpenstraße). There you take the bus line8 or 3 on the opposite side of the road. If you take the bus line 8, get off at the 6th busstop ”Zentrum-Ferdinand Hanusch Platz”, otherwise get off at ”Theatergasse”. At ”Theater-gasse” you first have to cross the river via the Staatsbrucke and then turn right to get to the busstop ”Zentrum-Ferdinand Hanusch Platz”, which is a two minute walk away from the Sternbrau.

Alternatively, it will take you about 30 minutes to walk. Just get to the main road, turn left(west) and follow the road until you have reached the bus stop ”Zentrum-Ferdinand HanuschPlatz”.

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Program

Thursday (May 4th, 2017)

12:00-13:00 Registration13:00-13:10 Opening13:10-13:30 Benjamin Stadlbauer

Brownian-dynamics simulations of nonopore protein sensing13:30-13:50 Gregor Mitscha-Baude

Simulation of nanopores with the Poisson-Nernst-Plack-Stokes and adaptivefinite elements

13:50-14:10 Stefan RiggerApproximation of Multisymmetric Functions

14:10-14:30 Gudmund PammerComputing Cubature Formulas for Multisymmetric Functions and Applica-tions to Stochastic Partial Differential Equations

14:30-14:50 Amirreza KhodadadianOptimal multi-level Monte Carlo method for the stochastic drift-diffusion-Poisson system

14:50-15:20 Coffee Break15:20-15:40 Boaz Blankrot

Multiple scattering approach for dielectric metamaterial analysis15:40-16:00 Lukas Kogler

ASC-AMG, a parallel AMG-solver for Netgen/NGSolve16:00-16:20 Bernd Schwarzenbacher

Algebraic Multigrid for Maxwell’s Equations16:20-16:40 Daniel Jodlbauer

Robust and Efficient Solvers for Fluid-Structure Interaction16:40-17:00 Markus Gasteiger

ADI type preconditioners for the steady state inhomogeneous Vlasov equa-tion

17:00-17:15 Break17:15-17:35 Gerhard Kitzler

A tensor product framework for kinetic equations17:35-17:55 Darian M. Onchis

Numerical considerations of consistency and stability in spline-type spaces17:55-18:15 Christian Gerhards

Modeling and Numerical Aspects of Inverse Poblems in Geomagnetism18:15-18:35 Markus Schobinger

Simulation of Eddy Currents in an Iron Ring Core Using a Multi-ScaleMethod

19:30 Joint Dinner at Sternbrau

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Friday (May 5th, 2017)

08:30-08:50 Mario Luiz Previatti de SouzaConvergence result for IRGNM type method under a tangential cone condi-tion in Banach space

08:50-09:10 Paolo Di StolfoDual weighted residual error estimation for the finite cell method

09:10-09:30 Bernhard EndtmayerAdaptive Mesh Refinement for Multiple Goal Functionals

09:30-09:50 Joscha GedickeResidual-based a posteriori error analysis for symmetric mixed Arnold-Winther FEM

09:50-10:10 Thomas FuhrerOn the DPG method for Signorini problems

10:10-10:40 Coffee Break10:40-11:00 Nina Ovcharova

Numerical methods for nonmonotone contact problems in continuum me-chanics

11:00-11:20 Markus FaustmannLocal convergence of the boundary element method on polyhedral domains

11:20-11:40 Thomas ApelSuperconvergent graded meshes: First results

11:40-12:00 Philip Lukas LedererPolynomial robust stability analysis for H(div)-conforming finite elements forthe Stokes equations

12:00-12:20 Olaf SteinbachRegularization error estimates for distributed control problems

12:20-13:20 Lunch Break13:20-13:40 Clemens Hofreither

A Black-Box Algorithm for Fast Matrix Assembly in Isogeometric Analysis13:40-14:00 Linus Wunderlich

Isogeometric mortar methods in solid mechanics14:00-14:20 Svetlana Matculevich

Functional a posteriori error estimates and adaptivity for IgA schemes14:20-14:40 Ioannis Toulopoulos

Time Discontinuous Galerkin Multipatch Isogeometric Analysis of ParabolicDiffusion Problems

14:40-15:10 Coffee Break15:10-15:30 Stefan Dohr

Space-time boundary element methods for the heat equation15:30-15:50 Marco Zank

Space-Time Boundary Element Method for the Wave Equation15:50-16:10 Bernhard Stiftner

Linear second order implicit-explicit time-integration of the (eddy-currents-)Landau-Lifshitz-Gilbert equation

16:10-16:30 Harald HofstatterTime propagators for Schrodinger-type equations with expensive-to-evaluatenonlinear part

16:30 Closing

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Brownian-dynamics simulations of nonopore protein sensing

Benjamin StadlbauerTU Wien, Wiedner Hauptstraße 8, 1040 Wien, benjamin.stadlbauer@tuwien.ac.at

Gregor Mitscha-BaudeTU Wien, Wiedner Hauptstraße 8, 1040 Wien, gregor.mitscha-baude@gmx.at

Clemens HeitzingerTU Wien, Wiedner Hauptstraße 8, 1040 Wien, clemens.heitzinger@tuwien.ac.at

Keywords: nanopores, Langevin simulations, Brownian dynamic simulations, stochasticdifferential equation

ABSTRACT

Nanopores are tiny holes in insulating membranes which connect two electrolyte chambers oneach side. If an electric potiential is applied, it induces ion transport through this opening,which is measured. If a target molecule moves inside the nanopore, it partially blocks thechannel and causes a reduction in the current trace. In this way, one can detect moleculessuch as DNA strands or proteins, where the amplitude, duration, and the shape of the eventsignal distinguishes between different types of particles. While nanopore DNA sensing is betterunderstood, protein sensing still needs to be studied better. This also implies challenges formodeling and numerical analysis. To simulate protein sensing with nanopores, approaches suchas molecular dynamics (MD) and simple Langevin simulations have been used. However, MDsimulations are computationally extremely expensive and therefore the physics are sometimessimplified in order to reduce runtime. Furthermore, simple Langevin simulations do not alwaysproduce realistic results because of the lack of important physical and chemical details.We present another model for modeling protein sensing, which is also based on the Langevinequation, a stochastic differential equation. In contrast, we also consider the contribution ofthe electroosmotic flow, which is computed with a continuum model [1]. In addition, we takenon-constant anisotropic diffusivity of the ions as well as of the proteins into account, andtherefore the results we obtain are much more realistic. In contrast to MD simulations, we canconsider long-time unspecific binding of the protein inside the nanopore. Additionally, becauseour simulations are much faster, it is possible to obtain hundreds or thousands of events andcompile statistics, which is necessary, since the movement of a target molecule is a stochasticprocess. Hence it has become possible to simulate the trajectories of the particles and thereforethe time-dependent event signals in various setups and furthermore to examine the distributionsof the dwell times of the events and also of the amplitudes of the current reductions.With our model we are not only able to reproduce experimental data, but also to answer openquestions regarding the interpretation of certain events which have been observed in very recentexperiments.

REFERENCES

[1] Gregor Mitscha-Baude, Andreas Buttinger-Kreuzhuber, Gerhard Tulzer, and ClemensHeitzinger. Adaptive and iterative methods for simulations of nanopores with the PNP-Stokes equations. Journal of Computational Physics, 338:452-476, 2017.

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Simulation of nanopores with the Poisson-Nernst-Plack-Stokesand adaptive finite elements

Gregor Mitscha-BaudeTU Wien, Wiedner Hauptstraße 8–10, gregor.mitscha-baude@gmx.at

Benjamin StadlbauerTU Wien, Wiedner Hauptstraße 8–10

Clemens HeitzingerTU Wien, Wiedner Hauptstraße 8–10

Keywords: nanopores, PDE models for transport, goal-oriented adaptivity

ABSTRACT

Nanopores are tiny holes which enable ions and biological molecules to flow through an otherwiseinsulating membrane. They can be used for detection of large molecules by monitoring theelectrical current through the pore, because these molecules block ion flow when they enter thenanopore and cause a measurable drop in current. Discriminating molecules by their currentsignatures has made it possible to sequence DNA. The sequencing of proteins with nanopores isstill an open problem, and would have huge scientific and medical implications. The phenomenaassociated with nanopore sensing offer a host of fascinating problems for modeling, analysisand scientific computing. We employ the Poisson and Nernst-Planck equations to model ioncurrent, the Stokes system to describe the flow of water around particles, and a Fokker-Planckequation to model the stochastic transport of proteins through a nanopore. All these equationsare coupled to each other, including nonlinear interactions in the PoissonNernst-Planck-Stokes(PNPS) system. First, we present our finite element approach to solve the PNPS equationson realistic 3D nanopore geometries [1]. Since the focus is on obtaining quantities of practicalinterest, namely the ion current and electrophoretic force induced on a protein, we develop agoal-oriented adaptive mesh refinement strategy. Second, we investigate non-constant diffusionconstants of ions and proteins. Diffusion constants enter both the PNPS and the Fokker-Planckequations. We find that they are reduced considerably inside a nanopore. To compute them,we again need to solve a Stokes equation for many locations of the ion/protein. In this way, weobtain a stack of three PDE models on top of each other, where a large number of solutions ofone PDE has to be evaluated to determine the coefficients of the next one: The Stokes equationdetermines the diffusion constant in the PNPS system, which in turn determines the force on aprotein in the Fokker-Planck model. This poses the not entirely trivial side question of how tointerpolate a coefficient from as few evaluations as possible in an arbitrary 3D geometry.

REFERENCES

[1] Gregor Mitscha-Baude, Andreas Buttinger-Kreuzhuber, Gerhard Tulzer, and ClemensHeitzinger. Adaptive and iterative methods for simulations of nanopores with thePNPStokes equations. Journal of Computational Physics, 338:452-476, 2017.

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Approximation of Multisymmetric Functions

Stefan RiggerTU Wien, A-1040 Vienna, Austria, stefan.rigger@student.tuwien.ac.at

Gudmund PammerTU Wien, A-1040 Vienna, Austria, e1225483@student.tuwien.ac.at

Clemens HeitzingerTU Wien, A-1040 Vienna, Austria, clemens.heitzinger@tuwien.ac.at

Keywords: quadrature and cubature formulas, multivariate integration,permutation-invariance, multisymmetric polynomials

ABSTRACT

Many interesting applications in physics exhibit symmetries that can be exploited to reducecomputational effort. Therefore, we consider a mathematical setting that reflects these symmetryproperties. We introduce the following notion of invariance under permutations: for a subgroupG of SN , we call an N -variate real-valued function G-invariant if f σ = f for every σ in G.We prove that the space of G-invariant continuous functions is a Banach space that containsthe space of G-invariant polynomials as a dense subspace. An analogous theorem can be shownin the case of spaces of p-integrable functions. We also show that the Taylor polynomials of aG-invariant function centered at a G-invariant point are G-invariant.These results naturally motivate the idea of cubature rules for G-invariant functions, where wedemand that a rule of order d should integrate every G-invariant polynomial of degree less thanor equal to d exactly. We prove error bounds for cubature formulas of this type. Furthermore, weinvestigate an important special case of G-invariance, namely multisymmetry groups. Finally,we prove that in a certain sense, there is no curse of dimensionality on spaces of multisymmetricpolynomials (in contrast to ordinary polynomial spaces).

REFERENCES

[1] C. Heitzinger, G. Pammer and S. Rigger. Cubature formulas for multisymmetric func-tions and applications to stochastic partial differential equations. Submitted, 2017.

[2] C. Heitzinger, G. Pammer and S. Rigger. Numerical solution of multisymmetric stochas-tic partial differential equations. Inverse Problems, In preparation, 2017.

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Computing Cubature Formulas for Multisymmetric Functionsand Applications to Stochastic Partial Differential Equations

Gudmund PammerTU Wien, A-1040 Vienna, Austria, e1225483@student.tuwien.ac.at

Stefan RiggerTU Wien, A-1040 Vienna, Austria, stefan.rigger@student.tuwien.ac.at

Clemens HeitzingerTU Wien, A-1040 Vienna, Austria, clemens.heitzinger@tuwien.ac.at

Keywords: quadrature and cubature formulas, multivariate integration,permutation-invariance, multisymmetric polynomials

ABSTRACT

Many interesting applications in physics that can be modeled by stochastic partial differentialequations exhibit symmetries that can be exploited to reduce computational effort. Especially inthe case of stochastic PDE, the numerical solution of such problems requires the numerical inte-gration of high-dimensional integrals, since the Curse of Dimensionality is encountered frequentlyand problems turn out to be computationally highly demanding. By exploiting permutation-invariance properties, the complexity of the integration problem can be significantly reduced,and problems that are unsolvable using a naive approach become tractable as in [3].We demonstrate how cubature formulas for multisymmetric functions can be calculated in prac-tice, making use of the special structure of multisymmetry groups. We compare cubature for-mulas for multisymmetric functions to tensor product rules in the low-dimensional case and toMonte-Carlo and sparse-grid methods in the high-dimensional case. Finally, we compute theexpectation of the solution of a stochastic partial differential equation both using a quasiMonte-Carlo method and our proposed formulas, discussing the benefits of our method. In the case ofsmooth multisymmetric functions, the results indicate that the proposed cubature formulas arehighly efficient.

REFERENCES

[1] C. Heitzinger, G. Pammer and S. Rigger. Cubature formulas for multisymmetric func-tions and applications to stochastic partial differential equations. Submitted, 2017.

[2] C. Heitzinger, G. Pammer and S. Rigger. Numerical solution of multisymmetric stochas-tic partial differential equations. In preparation, 2017.

[3] D. Nuyens, G. Suryanarayana and M. Weimar. Rank-1 lattice rules for multivariateintegration in spaces of permutation-invariant functions. Advances in ComputationalMathematics, 2016.

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Optimal multi-level Monte Carlo method for the stochastic drift-diffusion-Poisson system

Amirreza KhodadadianTU Wien, Wiedner Hauptstrasse 8–10, 1040, Wien, Amirreza.Khodadadian@TUWien.ac.at

Leila TaghizadehTU Wien, Wiedner Hauptstrasse 8–10, 1040, Wien, Leila.Taghizadeh@TUWien.ac.at

Clemens HeitzingerTU Wien, Wiedner Hauptstrasse 8–10, 1040, Wien, Clemens.Heitzinger@TUWien.ac.at

Keywords: Silicon nanowire sensors, multi-level Monte Carlo, stochasticdrift-diffusion-Poisson system

ABSTRACT

The stochastic drift-diffusion-Poisson system [1] serves as a leading example for the develop-ment of optimal numerical algorithms for systems of stochastic PDE. The model consists of thePoisson-Boltzmann equation to model the electrolyte and the drift-diffusion-Poisson system tomodel the charge transport in the transducer. Also, a reaction equation is applied to model theassociation of target molecules to the sensor surface and their dissociation.

The optimal multi-level Monte Carlo (MLMC) is developed to obtain an accurate estimation ofthe expected value of the solution of the system [1]. This allows to find the optimal choice ofdiscretization parameters. In other words, we minimized the overall computational cost for aprescribed total error i.e., spatial error (finite-element discretization) as well as statistical error(i.e., randomness of the system). Here, we define a global optimization problem which minimizesthe computational complexity such that the error bound is less or equal to a given tolerancelevel. To further improve the computational efficiency, a randomized low-discrepancy sequencesuch as a randomly shifted lattice are applied as well [2].

The applications considered here are noise and fluctuations in silicon nanowire sensors andmulti-gate transistors. The multi-level approach shows noticeable advantages compared to thesingle-level method, where for lower error bounds, the computational work is reduced by fourorders of magnitude. The speed-up becomes better as the error tolerance decreases.

REFERENCES

[2] Leila Taghizadeh, Amirreza Khodadadian, and Clemens Heitzinger. The optimal multi-level Monte-Carlo approximation of the stochastic drift-diffusion-Poisson system. Com-puter Methods in Applied Mechanics and Engineering (CMAME), 318 (2017): 739-761.

[1] Leila Taghizadeh, Amirreza Khodadadian, and Clemens Heitzinger. Optimal multi-levelrandomized quasi Monte-Carlo method for the stochastic drift-diffusion-Poisson system.pages 1–21. In preparation.

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Multiple scattering approach for dielectric metamaterial analysis

Boaz BlankrotInstitute for Analysis and Scientific Computing, TU Wien; Wiedner Hauptstrasse 8–10, A-1040Vienna, Austria; boaz.blankrot@tuwien.ac.at

Clemens HeitzingerInstitute for Analysis and Scientific Computing, TU Wien; Wiedner Hauptstrasse 8–10, A-1040Vienna, Austria; clemens.heitzinger@tuwien.ac.at

Keywords: Computational Electromagnetics, Integral Equation Methods, Julia

ABSTRACT

In this talk we consider electromagnetic scattering from a collection of many similar arbitrarily-shaped inclusions in the metamaterial regime (R/λ ≈ 0.5). We use Julia, a young high-performance programming language for numerical computing, to implement a solver for thisproblem in two dimensions. Generally, an integral-equation approach is appropriate for suchopen scattering problems, however this yields dense system matrices that are costly to solve.Additionally, moving or rotating the inclusions would require re-computation of a large portionof the system matrix. To address these issues, we combine the integral equation method with amultiple-scattering formulation [1], which replaces each inclusion with a scattering matrix thattranslates incoming multipole expansions to an outgoing one. In this approach the integral equa-tion is solved locally in the preprocessing stage, and only once for each type of inclusion. Forsmooth inclusions in our regime the multipole expansion length is substantially smaller than thenumber of points used to discretize the boundary. Therefore, the number of degrees of freedomin the global system is reduced by an order of magnitude. We accelerate our solver by meansof a fast multipole method, resulting in time complexity that scales roughly linearly with thenumber of inclusions.

We describe how this tool can be used in the design and analysis of two-dimensional dielectricmetamaterials. For example, we examine the Luneburg lens and its realization by means of agraded index photonic crystal [2]. In addition, the possibility of automating the design processby applying optimization techniques to this solver is discussed.

REFERENCES

[1] J. Lai, M. Kobayashi, and L. Greengard. A fast solver for multi-particle scattering in alayered medium. Optics Express, Vol. 22, 20481–20499, 2014.

[2] F. Gaufillet and E. Akmansoyd. Graded Photonic Crystals for Luneburg Lens. IEEEPhotonics Journal, Vol. 8, 1–11, 2016.

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ASC-AMG, a parallel AMG-solver for Netgen/NGSolve

Lukas KoglerVienna University of Technology, Vienna, lukas.kogler@tuwien.ac.at

Joachim SchoberlVienna University of Technology, Vienna, joachim.schoeberl@tuwien.ac.at

Keywords: AMG, MPI, FEM

ABSTRACT

We introduce ASC-AMG, which stands for ”Alternative Strong Connections AMG”, a MPI-parallel algebraic multigrid solver for Netgen/NGSolve which is currently under development.It can be seen as a variant of AMG by agglomeration where we take a new approach to definestrong connections between degrees of freedom.

In most traditional AMG solvers, the strength of connection between degrees of freedoms i andj is measured by the corresponding off-diagonal entry in the system matrix A. Our approachis more closely related to the underlying FEM-discretization. On the element matrix level, wecompute equivalent but simpler structured replacement element matrices which, when assem-bled, give us a replacement matrix A for the entire FEM-System such that ||u||2A ≈ ||u||2A =∑

ij wij(ui − uj)2, where wij are the weights we use to measure strength of connection.

We present our approach to parallel agglomeration based on the replacement matrix and showthe formalism we use to describe the parallel nature of degrees of freedom in an MPI-basedsetting.

We will also briefly show our approach to coarse grid interpolation, which also uses the replacement-matrix and allows for communication-less transfer between grid levels.

Finally, we will show scalability results on clusters up to 2000 cores.

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Algebraic Multigrid for Maxwell’s Equations

Joachim SchoberlTU Wien, Wiedner Hauptstrasse 8-10, 1040 Wien, joachim.schoeberl@tuwien.ac.at

Bernd SchwarzenbacherTU Wien, Wiedner Hauptstrasse 8-10, 1040 Wien,bernd.schwarzenbacher@student.tuwien.ac.at

Keywords: Maxwell’s equations, algebraic multigrid, finite element method

ABSTRACT

We present an algebraic multigrid method to solve large sparse systems of linear equations arisingfrom Nedelec finite element discretization of problems posed in H(curl,Ω) [1]. The main part isa new criterion, motivated by additive Schwarz theory, for a sensible mesh collapse algorithm toconstruct a hierarchy of coarse representations from the original fine mesh. The criterions aimis to preserve the homology of the domain.

Together with a prolongation operator based on the paper of Reitzinger-Schoberl [2], whichmaps fine curl-free functions to coarse curl-free functions, we achieve, that the De Rham complexremains a complete sequence on each level. This for one gives us robustness in parameter jumps.Further we can now use a Hiptmair smoother [3] to obtain robustness in small regularizationparameters.

The method was implemented within the finite element software NGSolve (ngsolve.org). Inthe end we show numerical results for magnetostatic examples in 3D demonstrating parallelscalability on shared memory computers.

REFERENCES

[1] Peter Monk. Finite Element Methods for Maxwell’s Equations, Oxford University PressInc. New York, 2003

[2] Reitzinger, S. and Schoberl, J. An algebraic multigrid method for finite element dis-cretizations with edge elements. Numerical Linear Algebra with Applications, Vol. 9,223–238, 2002.

[3] Ralf Hiptmair. Multigrid method for Maxwell’s equations. SIAM J. Numer. Anal., Vol.36 204–225, 1999.

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Robust and Efficient Solvers for Fluid-Structure Interaction

Daniel JodlbauerJohannes Kepler University Linz, Doctoral Program “Computational Mathematics”,Altenberger Straße 69, 4040 Linz, daniel.jodlbauer@dk-compmath.jku.at

Ulrich LangerInstitute of Computational Mathematics JKU Linz, Altenbergerstraße 69, 4040 Linz,ulanger@numa.uni-linz.ac.at

T. WickCentre de Mathematiques Appliquees (CMAP), Ecole Polytechnique, Route de Saclay, 91128PALAISEAU Cedex, France, thomas.wick@polytechnique.edu

Keywords: solver, monolithic, fluid-structure-interaction, ALE, parallelization

ABSTRACT

Fluid-structure-interaction problems have a wide range of applications, but their efficient solu-tion remains challenging. In this work we provide all details necessary for a monolithic ALEimplementation using the finite element library deal.II. To actually solve the arising linear sys-tems, we develop a preconditioner based on an approximate block-wise LU-factorization, split-ting the coupled system of equations into its natural fluid, solid and mesh sub-problems. Nu-merical results illustrate the robust convergence with respect to different material parametersand mesh-size h, and with an acceptable dependence on the time-step size ∆t. Furthermore,some preliminary results regarding parallelization are shown.

REFERENCES

[1] D. Jodlbauer, T. Wick. Monolithic FSI. Radon Series on Comp. App. Math, Vol. 20,2017.

[2] D. Jodlbauer. Robust Preconditioners for Fluid-Structure Interaction. Master thesis,2016

[3] U. Langer and H. Yang. Robust and efficient monolithic fluid-structure-interactionsolvers. Int. J. Numer. Meth. Engng., Vol. 108, 303–325, 2016.

[4] T. Wick. Solving Monolithic Fluid-Structure Interaction Problems in Arbitrary La-grangian Eulerian Coordinates with the deal.II Library. Archive of Numerical Software,Vol. 1, 1–19, 2013.

20

ADI type preconditioners for the steady state inhomogeneousVlasov equation

Markus GasteigerUniversity of Innsbruck, Technikerstr. 13, A-6020 Innsbruck, markus.gasteiger@uibk.ac.at

in collaboration with

Lukas EinkemmerUniversity of Innsbruck

Alexander OstermannUniversity of Innsbruck

David TskhakayaTechnical University of Vienna

Keywords: Vlasov equation, preconditioning, iterative methods, plasma physics.

ABSTRACT

The purpose of this work is to find numerical solutions of the steady state inhomogeneous Vlasovequation. This problem has a wide range of applications in the kinetic simulation of non-thermalplasmas. However, the direct application of either time stepping schemes or iterative methods(e.g. Krylov based methods or relaxation schemes) is computationally expensive. In the formercase the slowest timescale in the system forces us to perform a long time integration while inthe latter case a large number of iterations is required.

In this paper we propose a preconditioner based on an alternating direction implicit (ADI)type splitting method. This preconditioner is then used with both GMRES and Richardsoniteration. The resulting numerical schemes scale almost ideally (i.e. the computational effort isproportional to the number of grid points). Numerical simulations conducted show that thiscan result in a speedup of close to two orders of magnitude (even for intermediate grid sizes)with respect to the unpreconditioned case. In addition, we discuss the characteristics of thesenumerical methods and show the results for a number of numerical simulations.

REFERENCES

M. Gasteiger, L. Einkemmer, A. Ostermann, and D. Tskhakaya. Alternating direc-tion implicit type preconditioners for the steady state inhomogeneous Vlasov equation.Journal of Plasma Physics, 83(1), 2017.

21

A tensor product framework for kinetic equations

Gerhard KitzlerVienna UT, Wiedner Hauptstraße 8-10, 1030 Wien, gerhard.kitzler@tuwien.ac.at

Joachim SchoberlVienna UT, Wiedner Hauptstraße 8-10, 1030 Wien, joachim.schoeberl@tuwien.ac.at

Keywords:

ABSTRACT

In the talk we present a tensor product framework for the solution of higher dimensional prob-lems. We show efficient evaluation of bilinear forms and solution fields and that both can bereduced to the individual components of the tensor product. For that reason there is no needto implement high dimensional finite elements, only the tensor product structure needs to betreated properly. The bilinear forms for continuous as well as discontinuous Galerkin methodscan be specified within the Python interface NGS-Py of the finite element library NGSolve.Finally we demonstrate the usability of the framework by application to kinetic equations gov-erning carrier transport in semiconductors.

REFERENCES

[1] www.ngoslve.org.

22

Numerical considerations of consistency and stability in spline-type spaces

Darian M. OnchisUniversity of Vienna, Faculty of Mathematics, Oskar-Morgenstern-Platz 1,darian.onchis@univie.ac.at

Simone ZappalaUniversity of Vienna, Faculty of Mathematics, Oskar-Morgenstern-Platz 1,simone.zappala@univie.ac.at

Keywords: spline-type systems, signal decomposition, numerical realizations, stability andconsistency, Gabor system

ABSTRACT

Spline-type spaces are a class of shift-invariant spaces distinguished by the possession of a Rieszbasis which consists of a set of translates along some lattice of a finite family of atoms. Inthis spaces, we characterize generating sets and subgroups that ensure the invertibility of thesynthesis operator; we call this property consistency. Because the characterization of a splinetype space belongs to its representation in the Fourier domain, we established a theoretical andnumerical method to select proper modulations for an input function or signal. Motivated by thisselection in the frequency domain that can lower the computational complexity of traditionalframe decomposition such as Gabor frame, we continued the study of the stability of spline-type systems through the study of continuity in the Frobenius norm of the synthesis operatorunder deformation of the generating set in the Fourier domain. We tested numerically theresults for the construction of approximate dual Gabor-like frames. We present the advantagesof this approach in both flexibility and speed. The method allows in a natural way to handlenon standard Gabor constructions like non-uniformity in frequency and the reductions of thenumber of used modulations.

REFERENCES

[1] H. Feichtinger and D. Onchis. Constructive realization of dual systems for generatorsof multi-window spline-type spaces. Journal of computational and applied mathematics,Vol. 234.12, 3467–3479. 2010.

[2] K. Grochenig. Foundations of time-frequency analysis, Birkhauser Basel, 2001.

[3] A Ron and Z. Taylor. Generalized shift-invariant systems. Constructive approximation,Vol. 22.1, 1–45. 2005.

23

Modeling and Numerical Aspects of Inverse Poblems in Geomag-netism

Christian GerhardsUniversity of Vienna, Computational Science Center, Oskar-Morgenstern Platz 1, 1090 Wien

Keywords: Inverse Potential Field Problems, Approximation on the Sphere, Geomagnetism

ABSTRACT

Several satellite missions have provided and are providing continuous measurements of theEarth’s magnetic field over the last 15-20 years. A major task is the extraction of the differentcontributions of the measured field (e.g., the contribution due to dynamo effects in the Earth’score, the contribution stemming from magnetizations in the Earth’s crust, or the contributionproduced by tidal ocean flow).

After a brief introduction, this talk focuses on the separation of the core and the crustal contri-bution. That is, knowing the magnetic field B = Bcore + Bcrust on a sphere SR, is it possible toseparate the contributions Bcore and Bcrust? In general, this problem is non-unique. We providea modeling approach, based on the assumptions that Bcrust is generated by locally supportedmagnetizations on the spherical Earth’s surface, which yields uniqueness of the problem. Fur-thermore, we derive an optimization problem that leads to an approximation of Bcrust, givenonly the knowledge of B on SR. Eventually, we present first numerical examples based on radialbasis function expansions.

24

Simulation of Eddy Currents in an Iron Ring Core Using a Multi-Scale Method

Markus SchobingerTechnische Universitat Wien, Vienna, A-1040 Austria, markus.schoebinger@tuwien.ac.at

Joachim SchoberlTechnische Universitat Wien, Vienna, A-1040 Austria, joachim.schoeberl@tuwien.ac.at

Karl HollausTechnische Universitat Wien, Vienna, A-1040 Austria, karl.hollaus@tuwien.ac.at

Keywords: eddy current, multi-scale, nonlinear, network coupling

ABSTRACT

The nonlinear eddy current problem with network coupling - for a given voltage U find themagnetic vector potential A(t) ∈ H(curl) and current I(t) ∈ R so that∫

Ων(A) curlA curl v dΩ +

∂

∂t

∫ΩσAv dΩ =

∫Γ(Ω)

Kv dΓ

IR+

∫Γ(Ω)

∂A

∂tτ dΓ = U

for every v ∈ H(curl) - is solved to simulate the eddy currents in an iron ring core. In order toreduce the number of degrees of freedom, cylindrical coordinates are used to model the radiallysymmetric domain using only two dimensions. Furthermore the single laminates of the iron coreare not resolved in the mesh. Instead a multi-scale method is used to recover the local behavior.The quality of the simulation is checked using measurement data provided by the Institute ofElectrical Machines of the RWTH Aachen.

REFERENCES

[1] A. Bensoussan, J. Lions, and G. Papanicolaou. Asymptotic Analysis for Periodic Struc-tures. North-Holland, 2011.

[2] K. Hollaus and J. Schoberl. Homogenization of the eddy current problem in 2d. 14thInt. IGTE Symp., Graz, Austria, Sep. 2010, pp. 154–159.

25

Convergence result for IRGNM type method under a tangentialcone condition in Banach space

Barbara KaltenbacherAlpen-Adria Universitat Klagenfurt, 9020, barbara.kaltenbacher@aau.at

Mario Luiz Previatti de SouzaAlpen-Adria Universitat Klagenfurt, 9020, mario.previatti@aau.at

Keywords: Gauss Newton Method, Regularization, Discretization

ABSTRACT

This talk deals with a combined analysis of regularization and discretization of inverse prob-lems in Banach spaces, specifically in the context of partial differential equations (PDEs). Therelevant quantities - parameters and states - have to be discretized, e.g., by the finite elementmethod, and the error due to this discretization has to be appropriately estimated and controlledby error estimators and mesh refinement. Thus one of the main challenges is here to take intoaccount the interplay between mesh size, regularization parameter and data noise level. Thefocus on the PDEs setting is relevant to the adaptive discretization of the regularized prob-lems. Hence, I will present convergence of the Iteratively Regularized Gauss Newton Method(IRGNM) in its classical Tikhonov version and in the IRGM Ivanov version under a tangentialcone condition in Banach space setting. Convergence without source conditions has so far onlybeen proven under stronger restrictions on the nonlinearity of the operator and/or the spaces. Iwill also present how to obtain the a posteriori estimates and to achieve the prescribed accuracyby adaptive discretization using goal oriented error estimators. Such problems play a crucial rolein numerous applications, ranging from medical imaging via nondestructive testing (e.g. elec-trical impedancy tomography) to geophysical prospecting (e.g. inverse water ground filtration),with the Banach space setting assigned by the inherent regularity of the sought coefficients aswell as structural features such as sparsity.

REFERENCES

[1] B. Kaltenbacher and B. Hofmann. Convergence rates for the iteratively regularizedGaussNewton method in Banach spaces. Inverse Problems, Vol. 26 (2010) 035007

[2] B. Kaltenbacher, A. Kirchner, and S. Veljovi´c. Goal oriented adaptivity in the IRGNMfor parameter identification in PDEs: I. reduced formulation. Inverse Problems, Vol.30, (2014) 045001.

[3] B. Kaltenbacher, A. Neubauer, O. Scherzer. Iterative Regularization Methods for Non-linear Ill-Posed Problems. Walter de Gruyter, Berlin – New York, 2008.

26

Dual weighted residual error estimation for the finite cell method

Paolo Di StolfoDepartment of Mathematics, University of Salzburg, Hellbrunner Straße 34, 5020 Salzburg,Austriapaolo.distolfo@sbg.ac.at

Andreas RademacherFakultat fur Mathematik (LS X), Technische Universitat Dortmund, 44221 Dortmund,Germanyandreas.rademacher@tu-dortmund.de

Andreas SchroderDepartment of Mathematics, University of Salzburg, Hellbrunner Straße 34, 5020 Salzburg,Austriaandreas.schroeder@sbg.ac.at

Keywords: dual weighted residual method, finite cell method, mesh adaptivity

ABSTRACT

In this talk, we present a goal-oriented error control based on the dual weighted residual method(DWR) for the finite cell method (FCM), in which the computational domain is covered by asimpler enclosing domain [1]. The error identity resulting from the DWR approach allows for acombined treatment of the discretization and quadrature error introduced by the FCM. We usea localization technique based on a partition of unity proposed by Richter and Wick [2]. Wepresent an adaptive strategy with the aim to balance the two error contributions. Its performanceis demonstrated for linear problems in 2D with linear goal functionals.

REFERENCES

[1] J. Parvizian, A. Duster, E. Rank. Finite cell method. Computational Mechanics, Vo. 41(1), 121–133, 2007.

[2] T. Richter and T. Wick. Variational localizations of the dual weighted residual estimator.Journal of Computational and Applied Mathematics, Vol. 279, 192–208, 2015.

27

Adaptive Mesh Refinement for Multiple Goal Functionals

B. EndtmayerDoctoral Program Computational Mathematics, Johannes Kepler University, AltenbergerStraße 69, A-4040 Linz, Austria, bernhard.endtmayer@dk-compmath.jku.at

T. WickCentre de Mathematiques Appliquees (CMAP), Ecole Polytechnique, Route de Saclay, 91128PALAISEAU Cedex, France, thomas.wick@polytechnique.edu

Keywords: finite element method; mesh adaptivity; dual-weighted residual; partition-of-unity;multi-objective goal functionals; adjoint to the adjoint problem

ABSTRACT

In this presentation, we design a posteriori error estimation and mesh adaptivity for multiple goalfunctionals. Our method is based on a dual-weighted residual approach in which localizationis achieved in a variational form using a partition-of-unity. The key advantage is that themethod is simple to implement and backward integration by parts is not required. For treatingmultiple goal functionals we employ the adjoint to the adjoint problem (i.e., a discrete errorproblem) and suggest an alternative way for its computation. Our algorithmic developmentsare substantiated for elliptic problems in terms of four different numerical tests that covervarious types of challenges, such as singularities, different boundary conditions, and diverse goalfunctionals. Moreover, computations with higher-order finite elements are performed.

REFERENCES

[1] B. Endtmayer and T. Wick. A Partition-of-Unity Dual-Weighted Residual Approach forMulti-Objective Goal Functional Error Estimation Applied to Elliptic Problems. Com-putational Methods in Applied Mathematics, published online, doi:10.1515/cmam2017-0001, 2017.

[2] B. Endtmayer. Adaptive Mesh Refinement for Multible Goal Functionals. Master thesis,Institute of Computational Mathematics, JKU Linz, 2017.

28

Residual-based a posteriori error analysis for symmetric mixedArnold-Winther FEM

Carsten CarstensenHumboldt-Universitat zu Berlin, Unter den Linden 6, 10099 Berlin, Germany,cc@math.hu-berlin.de

Dietmar GallistlKarlsruher Institut fur Technologie, Englerstr. 2, 76131 Karlsruhe, Germany, gallistl@kit.edu

Joscha GedickeUniversitat Wien, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria, joscha.gedicke@univie.ac.at

Keywords: linear elasticity, mixed finite element method, a posteriori, symmetric stress finiteelements

ABSTRACT

This talk introduces an explicit residual-based a posteriori error analysis for the symmetricmixed finite element method in linear elasticity after Arnold-Winther with pointwise symmetricand H(div)-conforming stress approximation. Opposed to a previous publication, the residual-based a posteriori error estimator of this talk is reliable and efficient and truly explicit in thatit solely depends on the symmetric stress and does neither need any additional informationof some skew symmetric part of the gradient nor any efficient approximation thereof. Henceit is straightforward to implement an adaptive mesh-refining algorithm obligatory in practicalcomputations.

Numerical experiments verify the proven reliability and efficiency of the new a posteriori errorestimator and illustrate the improved convergence rate in comparison to uniform mesh-refining.A higher convergence rate for piecewise affine data is observed in the L2 stress error and repro-duced in non-smooth situations by the adaptive mesh-refining strategy.

29

On the DPG method for Signorini problems

Thomas FuhrerTU Wien, Wiedner Hauptstraße 8–10, 1040 Vienna, Austria,thomas.fuehrer@tuwien.ac.at

Norbert HeuerPontificia Universidad Catolica de Chile, Vickuna Mackenna 4860, Santiago, Chile,nheuer@mat.uc.cl

Ernst P. StephanLeibniz University Hannover, Welfengarten 1, 30167 Hannover, Germany,stephan@ifam.uni-hannover.de

Keywords: Contact problem, Signorini problem, variational inequality, DPG method, optimaltest functions, ultra-weak formulation

ABSTRACT

We derive and analyze discontinuous Petrov-Galerkin methods with optimal test functions forSignorini-type problems as a prototype of a variational inequality of the first kind. We presentdifferent symmetric and non-symmetric formulations where optimal test functions are only usedfor the PDE part of the problem, not the boundary conditions. For the symmetric case andlowest order approximations, we provide a simple a posteriori error estimate. In a second part, weapply our technique to the singularly perturbed case of reaction dominated diffusion. Numericalresults show the performance of our method and, in particular, its robustness in the singularlyperturbed case

REFERENCES

[1] T. Fuhrer, N. Heuer and E.P. Stephan. On the DPG method for Signorini problems.arXiv.org, Vol. arXiv:1609.00765, 2016.

30

Numerical methods for nonmonotone contact problems in con-tinuum mechanics

Nina OvcharovaUniversitat der Bundeswehr Munchen, Werner-Heisenberg-Weg 39, 85577 Neubiberg,Germany, nina.ovcharova@unibw.de

Keywords: non-monotone contact, regularization, non-smooth optimization, hp-adaptivity

ABSTRACT

We present several efficient numerical methods for non-convex, non-smooth variational prob-lems in non-monotone contact. Examples include non-monotone friction and adhesive contactproblems, delamination and crack propagation in adhesive bonding of composite structures. Achallenging problem is adhesive bonding in case of contamination. The nonsmoothness comesfrom the non-smooth data of the problems itself, in particular from non-monotone, multivaluedphysical laws involved in the boundary conditions. The variational formulation of the resultingboundary value problems leads to a class of non-smooth variational inequalities, the so-calledhemivariational inequalities (HVIs). The latter maybe viewed as a first order condition of a non-convex, non-smooth minimization problem. These problems are much harder to analyze andsolve than the classical variational inequality problems like Signorini contact or Tresca-frictionalproblems. The resulting HVI problem is first regularized and then discretized by either finiteelement or boundary element methods. In addition, we propose a novel regularized mixed for-mulation and provide a reliable a-posteriori error estimate enabling also hp-adaptivity. Anotherapproach to solve nonsmooth variational problems is by the strategy: first discretize by finite (orboundary) elements, then optimize using finite dimensional non-smooth optimization methodsas bundle or non-smooth trust region methods. Various numerical experiments illustrate thebehavior, the strength and the limitations of the proposed approximation schemes.

REFERENCES

[1] N. Ovcharova and L. Banz. Coupling regularization and adaptive hp-BEM for the so-lution of a delamination problem. Numerische mathematik, DOI: 10.1007/s00211-017-0879-5, 2017.

[2] M. Dao, J. Gwinner, D. Noll, and N. Ovcharova. Nonconvex bundle method with appli-cation to a delamination problem. Computational Optimization and Applications, Vol.65 (1), 173–203, 2016.

[3] J. Gwinner and N. Ovcharova. From solvability and approximation of variational in-equalities to solution of nondifferentiable optimization problems in contact mechanics.Optimization, Vol. 64 (8), 1683–1702, 2015.

31

Local convergence of the boundary element method on polyhe-dral domains

Markus FaustmannTU Wien, Institute for Analysis and Scientific Computing, Wiedner Hauptstraße 8-10, 1040Wien, markus.faustmann@tuwien.ac.at

Jens Markus MelenkTU Wien, Institute for Analysis and Scientific Computing, Wiedner Hauptstraße 8-10, 1040Wien, melenk@tuwien.ac.at

Keywords: boundary element method, local error, regularity

ABSTRACT

We analyze the local behavior of the boundary element method with quasi-uniform meshesfor Symm’s integral equation and the hyper-singular integral equation on polyhedral Lipschitzdomains. Globally, the rate of convergence of the error is limited by the regularity of the solution,which may be reduced due to singularities of the geometry or data. However, if the quantityof interest is only a subpart of the computational domain, we can hope for better convergencebehavior of the error.

In this talk, we provide sharp local a-priori estimates in stronger norms (L2 and H1) than theenergy norms. Thereby, the local error can be bounded by the local best-approximation errorand a global error in a weak norm.

Duality arguments are used to control the errors in the weak norm. They rely on elliptic shifttheorems that involve both the interior and exterior problems.

The numerical examples also confirm the sharpness of our estimates.

32

Superconvergent graded meshes: First results

Thomas ApelUniversitat der Bundeswehr Munchen, 85577 Neubiberg, Germany, thomas.apel@unibw.de

Mariano MateosUniversidad de Oviedo, Campus de Gijon, 33203 Gijon, Spain, mmateos@uniovi.es

Johannes PfeffererTU Munchen, Boltzmannstr. 3, 85748 Garching bei Munchen, Germany, pfefferer@ma.tum.de

Arnd RoschUniverstat Duisburg-Essen, Thea-Leymann-Straße 9, 45127 Essen, Germany,Arnd.Roesch@uni-due.de

Keywords: superconvergence mesh, graded mesh, error estimates

ABSTRACT

Superconvergent discretization error estimates can be obtained when the solution is smoothenough and the finite element meshes enjoy some structural properties. The simplest one is thatany two adjacent triangles form a parallelogram.

The solution of elliptic boundary value problems contains singularities in the vicinity of corners(and edges in 3D) leading to reduced convergence order in the case of quasi-uniform meshes. Aremedy is the use of graded meshes near these corners.

The question arises whether both approaches could be combined. The aim of the talk is topresent first results.

33

Polynomial robust stability analysis for H(div)-conforming finiteelements for the Stokes equations

Philip Lukas LedererInstitute for Analysis an Scientific Computing - TU Wien, Wiedner Hauptstraße 8-10, 1040Wien , philip.lederer@tuwien.ac.at

Joachim SchoberlInstitute for Analysis an Scientific Computing - TU Wien, Wiedner Hauptstraße 8-10, 1040Wien , joachim.schoeberl@tuwien.ac.at

Keywords: Navier Stokes equations, mixed finite element methods, discontinuous Galerkinmethods, high order methods

ABSTRACT

In this work we consider a discontinuous Galerkin method for the discretization of the Stokesproblem. We use H(div)-conforming finite elements (see [2]) as they provide major benefits suchas exact mass conservation and pressure-independent error estimates. The main aspect of thiswork lies in the analysis of high order approximations. We show that the considered method isuniformly stable with respect to the polynomial order k and provides optimal error estimates‖u− uh‖1,∗ + ‖Πp− ph‖0 ≤ c(h/k)s‖u‖s+1. To derive those estimates, we prove a krobust LBBcondition. This proof is based on a polynomial H2-stable extension operator. This extensionoperator itself is of interest for the numerical analysis of C0-continuous discontinuous Galerkinmethods for 4th order problems.

REFERENCES

[1] Lederer, P. and Schoberl, J.. Joachim Polynomial robust stability analysis for H(div)-conforming finite elements for the Stokes equations. arXiv preprint arXiv:1612.01482,2016

[2] Christoph Lehrenfeld and Joachim Schoberl. High order exactly divergence-free HybridDiscontinuous Galerkin Methods for unsteady incompressible flows Computer Methodsin Applied Mechanics and Engineering, Vol. 307, 339 – 361, 2016.

34

Regularization error estimates for distributed control problems

Martin NeumullerInstitut fur Numerische Mathematik, Johannes Kepler Universitat Linz, Altenberger Str. 69,4040 Linz, neumueller@numa.uni-linz.ac.at

Olaf SteinbachInstitut fur Numerische Mathematik, TU Graz, Steyrergasse 30, 8010 Graz,o.steinbach@tugraz.at

Keywords: Distributed control problem, regularization error

ABSTRACT

As a model problem we consider a distributed control problem with energy regularization inH−1(Ω). In the case of no control constraints the optimality system is reduced to a singularlyperturbed diffusion–reaction equation. This enables us to derive regularization error estimatesfor the optimal state u with respect to the target u. Depending on the regularity of u we obtaindifferent orders in the regularization parameter which is confirmed by numerical examples. Wealso discuss the case when the control is considered in L2(Ω).

REFERENCES

[1] M. Neumuller and O. Steinbach. Regularization error estimates for distributed controlproblems in energy spaces. Berichte aus dem Institut fur Numerische Mathematik, TUGraz, 2017.

35

A Black-Box Algorithm for Fast Matrix Assembly in IsogeometricAnalysis

Clemens HofreitherDepartment of Computational Mathematics, Johannes Kepler University Linz,Altenbergerstr. 69, 4040 Linz, Austria. <chofreither@numa.uni-linz.ac.at>

Keywords: Isogeometric Analysis, low-rank approximation, splines

ABSTRACT

We present a fast algorithm for assembling stiffness matrices in Isogeometric Analysis with tensorproduct spline spaces. The procedure exploits the facts that (a) such matrices have block-bandedstructure, and (b) they often have low Kronecker rank. Combined, these two properties allowus to reorder the nonzero entries of the stiffness matrix into a relatively small, dense matrix ortensor of low rank. A suitable black-box low-rank approximation algorithm is then applied tothis matrix or tensor. This allows us to approximate the nonzero entries of the stiffness matrixwhile explicitly computing only relatively few of them, leading to a fast assembly procedure.

The algorithm does not require any further knowledge of the used spline spaces, the geome-try transform, or the partial differential equation, and thus is black-box in nature. Existingassembling routines can be reused with minor modifications.

Numerical examples demonstrate significant speedups over a standard Gauss quadrature assem-bler for several geometries in two and three dimensions. The runtime scales sublinearly with thenumber of degrees of freedom in a large pre-asymptotic regime.

36

Isogeometric mortar methods in solid mechanics∗

Barbara WohlmuthZentrum Mathematik, Technische Universitat Munchen, Boltzmannstr. 3, 85748 Garching,wohlmuth@ma.tum.de

Linus WunderlichZentrum Mathematik, Technische Universitat Munchen, Boltzmannstr. 3, 85748 Garching,linus.wunderlich@ma.tum.de

∗ Funded within the DFG priority programme SPP 1748 and joint work with S. Reese, W.A.Wall and C. Wieners.

Keywords: biorthogonal basis functions, isogeometric mortar methods, vibroacoustics

ABSTRACT

The handling of multi-patch geometries is a key ingredient to practical applications of isogeo-metric analysis [2], as complicated domains can often not be reasonably represented by a singleNURBS patch. Weak couplings of isogeometric mortar patches [1] allow for a flexible and accu-rate coupling of patches, without requiring the tensor product meshes to match at the interface.We highlight several solid mechanical applications, one of them is the vibroacoustical analysis ofa violin bridge. The thin wooden component of a violin has an important influence on the acous-tics. The complicated and curved domain motivates the use of isogeometric mortar methods. Inaddition to the nine orthotropic material parameters, the thickness of the geometry is includedas an extra parameter. The resulting eigenvalue problem with ten parameters is solved in amulti-query context to handle the uncertainty in the material parameter. An efficient solutionin this context is guaranteed by the use of reduced basis techniques for eigenvalue problems [3].Biorthogonal basis functions were applied in [4] for weak patch coupling as well as isogeometricdiscretizations of contact problems. We extend these results by constructing biorthogonal basisfunctions with higher order approximation properties.

REFERENCES

[1] E. Brivadis, A. Buffa, B. Wohlmuth and L. Wunderlich. Isogeometric mortar methods.Comput. Methods Appl. Mech. Eng., Vol. 284, 292–319, 2014.

[2] J.A. Cottrell, T.J.R. Hughes and Y. Bazilevs. Isogeometric Analysis, Wiley, 2009.

[3] T. Horger, B. Wohlmuth and L. Wunderlich. Reduced basis isogeometric mortar ap-proximations for eigenvalue problems in vibroacoustics. Accepted for publication inMOREPAS, 2015.

[4] A. Seitz, P. Farah, J. Kremheller, B. Wohlmuth, W.A. Wall, A. Popp. Isogeometric dualmortar methods for computational contact mechanics. Comput. Methods Appl. Mech.Eng., Vol. 301, 259–280, 2016.

37

Functional a posteriori error estimates and adaptivity for IgAschemes

U. LangerRICAM, Austrian Academy of Sciences, Austria, Altenberger Straße 69 4040 Linz,ulrich.langer@ricam.oeaw.ac.at

S. MatculevichRICAM, Austrian Academy of Sciences, Austria, Altenberger Straße 69 4040 Linz,svetlana.matculevich@ricam.oeaw.ac.at

S. RepinSt. Petersburg V.A. Steklov Institute of Mathematics, Russia, Fontanka river embankment, 27,Sankt-Peterburg, Russia, 191011, repin@pdmi.ras.ru

Keywords: adaptivity for space-time IgA scheme, parabolic equation, functional type aposteriori error estimates

ABSTRACT

We are concerned with guaranteed error control of Isogeometric Analysis (IgA) numerical ap-proximations of elliptic boundary value problems (BVPs). The approach is discussed within theparadigm of classical linear Poisson Dirichlet model problem: find u : Ω→ Rd such that

−∆xu = f in Ω, u = uD on ∂Ω, (0.1)

where Ω ⊂ Rd, d ∈ 1, 2, 3, denotes a bounded domain having a Lipschitz boundary ∂Ω, ∆x isthe Laplace operator in space, f ∈ L2(Ω) is a given source function, and uD ∈ H1

0 (Σ) is a givenload on the boundary.

We conduct the numerical study of the functional a posteriori error estimates integrated intothe IgA framework. These so-called majorants and minorants were originally introduced in[1] and later applied to different mathematical models. Initially, the functional approach tothe error estimation in combination with IgA approximations (generated by tensor-productsplines) was investigated in [2] for (0.1). In the current work, we test the algorithm of themajorant reconstruction suggested [2], which allows the considerable reduction of the time-costsfor the error estimates calculation and, at the same time, generates guaranteed, sharp, andfully computable bounds of errors. Moreover, we combine functional error estimates with THB-Splines (the implementation provided by G+smo) and demonstrate their efficiency with respectto adaptive mesh generation in IgA schemes.

REFERENCES

[1] S. Repin. A posteriori error estimation for nonlinear variational problems by dualitytheory. Zapiski Nauch. Sem. V. A. Steklov Math. Institute in St.-Petersburg (POMI),Vol. 243, 201–214, 1997.

[2] S. K. Kleiss and S. K. Tomar. Guaranteed and sharp a posteriori error estimates inisogeometric analysis. Computers & Mathematics with Applications, Vol. 70(3), 167–190, 2015.

38

Time Discontinuous Galerkin Multipatch Isogeometric Analysisof Parabolic Diffusion Problems

Ioannis ToulopoulosJohann Radon Institute for Computational and Applied Mathematics (RICAM),Austrian Academy of SciencesAltenbergerstr. 69, A-4040 Linz, Austria

Keywords: Parabolic initial-boundary value problems, Discontinuous Galerkin methods,Space-time Isogeometric Analysis, A priori discretization error estimates, parallel solvers.

ABSTRACT

In this talk, we present a discontinuous Galerkin (dG) time multipatch Isogeometric (IgA)scheme for the numerical solution of linear parabolic problems. We derive the weak formulationby multiplying the Partial Differential Equation (PDE) by a test function depending on spatialand time variable, and then applying integration by parts in both variables. The resultingformulation helps on deriving the analogous discrete space-time dG-IgA form. We show that,the discrete bilinear form is elliptic on the IgA space with respect to a mesh-dependent energynorm. This property together with a corresponding boundedness property, consistency andapproximation results for the IgA spaces yields a priori discretization error estimates. Wepresent numerical results confirming the efficiency of the space-time method and the theoreticalerror estimates.This work was supported by the Austrian Science Fund (FWF) under the grant NFN S117-03and is based on [1].

REFERENCES

[1] C. Hofer, U. Langer, M. Neumuller, and I. Toulopoulos. Multipatch time discon- tin-uous Galerkin space-time isogeometric analysis of parabolic evolution problems, underpreparation, 2017.

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Space-time boundary element methods for the heat equation

Stefan DohrTU Graz, Institut fur Numerische Mathematik, Steyrergasse 30, 8010 Graz, Austria,dohr@math.tugraz.at

Olaf SteinbachTU Graz, Institut fur Numerische Mathematik, Steyrergasse 30, 8010 Graz, Austria,o.steinbach@tugraz.at

Keywords: Space-time boundary element methods, preconditioning, heat equation

ABSTRACT

In this talk we describe the boundary element method for the discretization of the timedepen-dent heat equation. In contrast to standard time-stepping schemes we consider an arbitrarydecomposition of the boundary of the space-time cylinder into boundary elements, which areline segments in temporal direction in the one-dimensional case n = 1, triangles in the twodi-mensional case n = 2, and tetrahedra in the three-dimensional case n = 3. Besides adaptiverefinement strategies this approach allows us to parallelize the computation of the global solutionof the whole space-time system. In addition to the derivation of boundary element methods forthe Dirichlet initial boundary value problem we state convergence properties and error estimatesof the approximations. Those estimates are based on the approximation properties of boundaryelement spaces in anisotropic Sobolov spaces. The systems of linear equations are solved withthe GMRES method. Based on the mapping properties of the single layer and the hypersingu-lar boundary integral operator we construct a preconditioner for the discretization of the firstboundary integral equation. The theoretical results are confirmed by numerical tests.

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Space-Time Boundary Element Method for the Wave Equation

Marco ZankInstitut fur Numerische Mathematik, Steyrergasse 30/III, 8010 Graz, zank@math.tugraz.at

Keywords: Wave Equation, Boundary Element Method, Space-Time Method

ABSTRACT

For the discretisation of the wave equation by boundary element methods the starting pointis the so-called Kirchhoff’s formula, which is a representation formula by means of boundarypotentials. In this talk different approaches to derive weak formulations of related boundaryintegral equations are considered. First, a brief overview of the Laplace transform method withboundedness and coercivity estimates in appropriate Sobolev spaces is given. Second, a space-time energetic formulation is motivated and discussed. For this space-time energetic formulationa space-time boundary element method is introduced and to derive an adaptive scheme an aposteriori error estimator based on the representation formula is used.

Finally, numerical examples for a one-dimensional spatial domain are presented and discussed.

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Linear second order implicit-explicit time-integrationof the (eddy-currents-)Landau-Lifshitz-Gilbert equation

Carl-Martin PfeilerTU Wien, Wiedner Hauptstraße 8-10, 1040 Wien, carl-martin.pfeiler@tuwien.ac.at

Dirk PraetoriusTU Wien, Wiedner Hauptstraße 8-10, 1040 Wien, dirk.praetorius@tuwien.ac.at

Michele RuggeriTU Wien, Wiedner Hauptstraße 8-10, 1040 Wien, michele.ruggeri@tuwien.ac.at

Bernhard StiftnerTU Wien, Wiedner Hauptstraße 8-10, 1040 Wien, bernhard.stiftner@tuwien.ac.at

Keywords: micromagnetism, finite elements, time-marching scheme, unconditionalconvergence.

ABSTRACT

Combining ideas from [1] and [2], we present a numerical integrator for the integration of theLandau-Lifschitz Gilbert equation which is unconditionally convergent and formally (almost)second order in time, but requires only the solution of one linear system per time-step. Moreover,only the exchange contribution is treated implicitly in time, while the lower-order contributionslike the computationally expensive stray field are treated explicitly in time. Moreover, we extendthe scheme to the coupling of the Landau-Lifschitz Gilbert equation with eddy-currents. Unlikeexisting integrators for this PDE system, the new integrator is unconditionally convergent and(almost) second order in time and requires only the solution of two linear systems per time-step.

REFERENCES

[1] F. Alouges, E. Kritsikis, J. Steiner and J.-C. Toussaint. A convergent and precise finiteelement scheme for Landau-Lifschitz-Gilbert equation. Numer. Math. 128, Vol. 3,407–430, 2014.

[2] D. Praetorius, M. Ruggeri and B. Stiftner. Convergence of an implicit-explicit midpointscheme for computational micromagnetics. arXiv:1611.02465, 2016.

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Time propagators for Schrodinger-type equations with expensive-to-evaluate nonlinear part

Harald HofstatterTechnische Universitat Wien, Institut fur Theoretische Physik, Wiedner Hauptstraße 8–10,A-1040 Wien, hofi@harald-hofstaetter.at

Othmar KochUniversitat Wien, Institut fur Mathematik, Oskar-Morgenstern Platz 1, A-1090 Wien,othmar@othmar-koch.org

Keywords: Time-dependent Schrodinger equation, Multi-Configuration Method, ExponentialMultistep–Lawson Method

ABSTRACT

We give an overview of numerical time integration methods for time-dependent nonlinear equa-tions

i∂tψ(t) = Aψ(t) +B(t, ψ(t))

of Schrodinger type, where the linear operator A is built up from (discretized versions of) Lapla-cians, and the nonlinear time-dependent operator B is well-behaved but expensive to evaluate.As an example we consider the equations appearing in the context of multi-configuration time-dependent Hartree–Fock (MCTDHF) calculations. We compare a number of established ap-proaches, comprising splitting methods, composition methods, exponential Runge–Kutta meth-ods, and exponential multistep methods. It is found that exponential predictor-corrector multi-step methods of Lawson type are particularly attractive in terms of accuracy, stability, andcomputational effort.

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List of participants

Speakers

Thomas Apel, p. 33UniBw MunichGermanythomas.apel@unibw.de

Boaz Blankrot, p. 17Vienna University of TechnologyAustriaboaz.blankrot@tuwien.ac.at

Paolo Di Stolfo, p. 27University of SalzburgAustriapaolo.distolfo@sbg.ac.at

Stefan Dohr, p. 40Graz University of TechnologyAustriadohr@math.tugraz.at

Bernhard Endtmayer, p. 28JKU LinzAustriabernhard.endtmayer@dk-compmath.jku.at

Markus Faustmann, p. 32Vienna University of TechnologyAustriamarkus.faustmann@tuwien.ac.at

Thomas Fuhrer, p. 30Vienna University of TechnologyAustriathomas.fuehrer@tuwien.ac.at

Markus Gasteiger, p. 21University of InnsbruckAustriamarkus.gasteiger@uibk.ac.at

Joscha Gedicke, p. 29University of InnsbruckAustriamarkus.gasteiger@uibk.ac.at

Christian Gerhards, p. 24University of ViennaAustriachristian.gerhards@univie.ac.at

Clemens Hofreither, p. 36JKU LinzAustriachofreither@numa.uni-linz.ac.at

Harald Hofstatter, p. 43Vienna University of TechnologyAustriahofi@harald-hofstaetter.at

Daniel Jodlbauer, p. 20JKU LinzAustriadaniel.jodlbauer@dk-compmath.jku.at

Amirreza Khodadadian, p. 16Vienna University of TechnologyAustriaamirreza.khodadadian@tuwien.ac.at

Gerhard Kitzler, p. 22Vienna University of TechnologyAustriagerhard.kitzler@tuwien.ac.at

Lukas Kogler, p. 18Vienna University of TechnologyAustrialukas.kogler@tuwien.ac.at

Philip Lukas Lederer, p. 34Vienna University of TechnologyAustriaphilip.lederer@tuwien.ac.at

Mario Luiz Previatti de Souza, p. 26University of KlagenfurtAustriamario.previatti@aau.at

Svetlana Matculevich, p. 38RICAM LinzAustriasvetlana.matculevich@ricam.oeaw.ac.at

Gregor Mitscha-Baude, p. 13Vienna University of TechnologyAustriagregor.mitscha-baude@gmx.at

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Darian Onchis, p. 23University of ViennaAustriadarian.onchis@univie.ac.at

Nina Ovcharova, p. 31UniBw MunichGermanynina.ovcharova@unibw.de

Gudmund Pammer, p. 15Vienna University of TechnologyAustriagudmund.pammer@gmail.com

Stefan Rigger, p. 14Vienna University of TechnologyAustriastefan.rigger@student.tuwien.ac.at

Markus Schobinger, p. 25Vienna University of TechnologyAustriamarkus.schoebinger@tuwien.ac.at

Bernd Schwarzenbacher, p. 19Vienna University of TechnologyAustriae1326919@student.tuwien.ac.at

Benjamin Stadlbauer, p. 12Vienna University of TechnologyAustriabenjamin.stadlbauer@tuwien.ac.at

Olaf Steinbach, p. 35Graz University of TechnologyAustriao.steinbach@tugraz.at

Bernhard Stiftner, p. 42Vienna University of TechnologyAustriabernhard.stiftner@tuwien.ac.at

Ioannis Toulopoulos, p. 39RICAM LinzAustriaioannis.toulopoulos@oeaw.ac.at

Linus Wunderlich, p. 37Munich University of TechnologyGermanylinus.wunderlich@ma.tum.de

Marco Zank, p. 41Graz University of TechnologyAustriazank@math.tugraz.at

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Further participants

Emmanuel AkinlabiUniversity of WarsawPolandemmanuel.akinlabi@fuw.edu.pl

Niklas AngleitnerVienna University of TechnologyAustrianiklas.angleitner@tuwien.ac.at

Winfried AuzingerVienna University of TechnologyAustriaw.auzinger@tuwien.ac.at

Lothar BanzUniversity of SalzburgAustriaLothar.Banz@sbg.ac.at

Andreas ByfutUniversity of SalzburgAustriaandreas.byfut@sbg.ac.at

Scott CongreveUniversity of ViennaAustriascott.congreve@univie.ac.at

Giovanni Di FrattaVienna University of TechnologyAustriagiovanni.difratta@gmail.com

Kirian DopfnerVienna University of TechnologyAustriakirian.doepfner@tuwien.ac.at

Lukas EinkemmerUniversity of InnsbruckAustrialukas.einkemmer@uibk.ac.at

Gregor GantnerVienna University of TechnologyAustriagregor.gantner@tuwien.ac.at

Alexander HaberlVienna University of TechnologyAustriaahaberl@asc.tuwien.ac.at

Matthias HochstegerVienna University of TechnologyAustriamatthias.hochsteger@tuwien.ac.at

Othmar KochUniversity of ViennaAustriaothmar@othmar-koch.org

Raoul KutilUniversity of SalzburgAustriakutilra@stud.sbg.ac.at

Gregor MilicicUniversity of SalzburgAustriagregor.milicic@sbg.ac.at

Michael NeunteufelVienna University of TechnologyAustriae1241601@student.tuwien.ac.at

Gunther OfGraz University of TechnologyAustriaof@tugraz.at

Maryam ParviziVienna University of TechnologyAustriam.parvizi2010@gmail.com

Ilaria PerugiaUniversity of ViennaAustriailaria.perugia@univie.ac.at

Jan PetscheUniversity of SalzburgAustriajan.petsche@sbg.ac.at

Alexander PichlerUniversity of ViennaAustriaalex.pichler@univie.ac.at

Mirko ResidoriUniversity of InnsbruckAustriamirko.residori@uibk.ac.at

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Alexander RiederVienna University of TechnologyAustriaalexander.rieder@tuwien.ac.at

Claudio RojikVienna University of TechnologyAustriaclaudio.rojik@tuwien.ac.at

Maximilian SamsingerUniversity of InnsbruckAustriaMaximilian.Samsinger@uibk.ac.at

Stefan SchimankoVienna University of TechnologyAustriastefan.schimanko@tuwien.ac.at

Joachim SchoberlVienna University of TechnologyAustriajoachim.schoeberl@tuwien.ac.at

Andreas SchroderUniversity of SalzburgAustriaandreas.schroeder@sbg.ac.at

Leila TaghizadehVienna University of TechnologyAustriaLeila.Taghizadeh@TUWien.ac.at

Markus WessVienna University of TechnologyAustriamarkus.wess@tuwien.ac.at

Christoph WintersteigerVienna University of TechnologyAustriachristoph.wintersteiger@tuwien.ac.at

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