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Workshop on

Wave Propagation and Scattering, Inverse Problems, and Applications in

Energy and the Environment

November 21-25, 2011

as part of the

Radon Special Semester 2011 on

Multiscale Simulation & Analysis in Energy and the Environment

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The efficient computation of wave propagation and scattering is a core problem in numerical mathematics, which is currently of great research interest and is central to many applications in energy and the envi- ronment. Two generic applications which resonate strongly with the central aims of this special semester are forward wave propagation in heterogeneous media and seismic inversion for subsurface imaging. As an example of the first application, modelling of absorption and scattering of radiation by clouds, aerosol and precipitation is used as a tool for interpretation of (e.g.) solar, infrared and radar measurements, and as a component in larger weather/climate prediction models in numerical weather forecasting. One key numerical component in this modelling is the prediction of the total optical properties and the full scattering matrix from an ensemble of irregular particles. The underlying mathematical problem is that of accurately computing high frequency wave propagation in a highly heterogeneous medium. As an example of the second application, inverse problems in wave propagation in heterogeneous media arise in the problem of imaging the subsurface below land or marine deposits. Solutions to this problem have a number of environmental uses, for example in the location of hydrocarbon-bearing rocks, in the mon- itoring of pollution in groundwater or in earthquake modelling. A seismic source is directed into the ground and the material properties of the subsurface are inferred by analysing the observed scattered field, recorded by sensors. The inversion process (a large scale optimisation problem) is complicated by the presence of multiple reflections and the fact that the scales involved in the exploration of sub-marine, sub-basalt or sub-salt oil reservoirs can be many kilometres in extent, leading to a challenging multi-scale problem. Current iterative methods for solving the inverse problem involve repeated solution of the for- ward problem (the computational kernel), which is typically a frequency-domain reduction of the elastic (or scalar) wave equation with high frequency and typically highly spatially varying wave speed. If the inversion is to be competitive, the key underlying problem to be overcome is the design of robust and scalable solvers for the large highly indefinite linear systems arising from these problems.

The workshop will bring together key numerical mathematicians whose interest is in the analysis and computation of wave propagation and scattering problems, and in inverse problems, together with prac- titioners from engineering and industry whose interest is in the applications of these core problems.

Particular problems to be considered will be (i) The design of accurate methods for solving frequency domain problems; (ii) The use of wave enriched and other hybrid approximation strategies in the solution of high-frequency problems; (iii) Fast linear algebra solvers for frequency domain problems in heteroge- neous media; (iv) advanced inverse problem approaches for wave problems such as reverse time migration which move away from traditional ray-based approaches in the high frequency case.

Workshop Organizers

Ivan G. Graham, University of Bath, UK

Ulrich Langer, Johann Radon Institute & University of Linz, Austria Jens Markus Melenk, Vienna University of Technology, Austria Mourad Sini, Johann Radon Institute, Austria

cover picture of tsunami simulation in Indian Ocean:

courtesy of J¨orn Behrens (KlimaCampus, Universit¨at Hamburg) and Widodo S. Pranovo (Indonesia).

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Welcome

to Linz and thank you very much for participating in the sixth RICAM Special Semester on Multi- scale Simulation & Analysis in Energy and the Environment, hosted by the Johann Radon Insitute for Computational and Applied Mathematics (RICAM) from October 3 to December 16, 2011.

Technological advances have greatly improved our quality of life. However, they bring with them a huge surge in energy requirements which in turn puts at risk our entire bio-sphere. It is of paramount importance to predict these risks and to develop better solutions for the future. One of the central tasks is the accurate simulation of multiphase flow above and under ground. The risk analysis and uncertainty quantification, as well as the assimilation of data require statistical tools and efficient solvers for stochastic and deterministic PDEs as well as for the associated inverse problems. The key features that make it extremely hard to predict these physical phenomena accurately are the multiple time and length scales that arise, as well as the lack of and uncertainty in data. Because of the highly varying scales involved, the resolution of all scales is currently impossible even on the largest supercomputers. While there is a fairly long history of empirically successful robust computational techniques for certain multiscale problems, the rigorous (numerical) analysis of such methods is of extremely high current interest.

The goal of the special semester is to provide a stimulating environment for civil engineers, hydrologists, meteorologists and other environmental scientists to address together with mathematicians working at the cutting edge of rigorous numerical analysis for multiscale (direct and inverse) problems the emerging challenges in the quantitative assessment of the risks and uncertainties of atmospheric and subsurface flow, focusing in particular on

• Simulation of Flow in Porous Media and Applications in Waste Management and CO2Sequestration

• Large-Scale Inverse Problems and Applications in the Earth Sciences

• Data Assimilation and Multiscale Simulation in Atmospheric Flow

• Wave Propagation and Scattering, Direct and Inverse Problems and Applications in Energy and the Environment

• Multiscale Numerical Methods and their Analysis and Applications in Energy and the Environment

• Stochastic Modelling of Uncertainty and Numerical Methods for Stochastic PDEs Specific activities planned for the Special Semester are

• 4 thematic workshops addressing some of the key topics of the Special Semester;

• Special Lecture Series on ”Multilevel Methods for Multiscale Problems”;

• Graduate Seminar on ”Multiscale Discretization Techniques”;

• Wednesday Research Kitchen;

• Public Lecture byProf. J¨orn Behrens (KlimaCampus, Universit¨at Hamburg) on

“Tsunami Fr¨uh-Warnung: Mathematik und Wissenschaftliches Rechnen im Dienste der Sicherheit”.

We sincerely hope that you enjoy your stay in Linz!

Local Organizing Committee Program Committee

Robert Scheichl, Bath & RICAM (Chair) Peter Bastian, University of Heidelberg, Germany J¨org Willems, RICAM (Coordinator) Mike Cullen, Met Office, Exeter, UK

Johannes Kraus, RICAM (Co-Coordinator) Heinz Engl, RICAM & University of Vienna, Austria Erwin Karer, RICAM (Co-Coordinator) Melina Freitag, University of Bath, UK

Ivan G. Graham, University of Bath, UK

Ulrich Langer, RICAM & University of Linz, Austria Markus Melenk, TU Vienna, Austria

Robert Scheichl, University of Bath, UK (Chair) Mary F. Wheeler, University of Texas at Austin, USA

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Contents

Information 2

Workshop Information . . . 2

Social Events . . . 2

Restaurants and Cafes . . . 2

General Information . . . 2

Program 5

Posters 7

Abstracts 8

Abstracts for Posters 17

List of Participants 20

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Information

Workshop Information

Registration. The workshop registration will be on November 21st, 2011 from 9:00 - 9:40 am next to the seminar room SP2 416 on the 4th floor of the Science Park Building 2 (see floor plan). Participants that arrive later in the week can register at the special semester office SP2 456.

Registration Fee. Non-invited participants are kindly asked to pay the registration fee in cash upon registration.

Campus plan and overview map as well as a floor plan of the 4th floor of the workshop venue (Sci- ence Park Building 2) are located on the next pages.

Seminar room. The workshop will take place in seminar room SP2 416 on the 4th floor of the Science Park Building 2 (see floor plan).

Program. A time schedule for the workshop is located on the backside of this booklet.

Coffee breaks. The coffee breaks will be in the corridor of the 4th floor of the Science Park Building 2.

Internet access. There will be an extra information sheet regarding internet access available at regis- tration.

Social Events

Welcome Reception & Poster Session. Monday, November 21st, 2011, 5:15 pm, on the 4th floor of the Science Park Building 2.

Conference Dinner. Thursday, November 24th, 2011, 7:00 pm, at the restaurant “Kepler’s”, situated in the Mensa building

Restaurants and Cafes

• Mensa Markt (lunch time only) - Main canteen of the University (see campus plan)

• KHG Mensa (lunch time only) - Smaller canteen - good traditional food (see overview map: “KHG Linz”)

• Pizzeria “Bella Casa” - Italian and Greek restaurant (located next to the tram stop)

• Chinese restaurant “Jadegarten” - (located close by the tram stop, adjacent to “Bella Casa”)

• Asia restaurant “A2” - (located behind the Science Park on Altenbergerstrasse)

• “Chat” cafe - coffee, drinks and sandwiches (located in the “H¨orsaaltrakt” - see overview map)

• Cafe “Sassi” - coffee, drinks and small snacks (located in the building “Johannes Kepler Universit¨at”

- see overview map)

• Bakery “Kandur” - bakery and small cafe (located opposite the tram stop)

General Information

Accommodation. The arranged accomodation for invited participants is the “Sommerhaus” hotel. You can find its location in the overview map on page 4.

Special Semester Office: Room SP2 456. The special semester administrator is Susanne Dujardin.

Audiovisual & Computer Support. Room SP2 458, Wolfgang Forsthuber or Florian Tischler.

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Orientation/ Local Transport. From the railway station you have to take tram number 1 or 2 in direction “Universit¨at”. It takes about 25 minutes to reach the desired end stop “Universit¨at”.

In order to get to the city center of Linz (“Hauptplatz”) and back you have to take again tram number 1 or 2 (about 20 minutes). For more information seewww.ricam.oeaw.ac.at/location/.

Taxi Numbers.

+43 732 6969 Ober¨osterreichische Taxigenossenschaft +43 732 2244 2244 Linzer Taxi

+43 732 781463 Enzendorfer Taxi & Transport +43 732 2214 Linzer Taxi

+43 732 660217 LINTAX TaxibetriebsgesmbH Further important phone numbers.

+43 (0)732 2468 5222 RICAM & Special Semester Office (Susanne Dujardin) +43 (0)732 2468 5250/5255 RICAM IT Support (Florian Tischler/ Wolfgang Forsthuber) +43 (0)732 2457-0 Reception of Hotel Sommerhaus

133 General emergency number for the police

144 General emergency number for the ambulance

More information about RICAM can be found at www.ricam.oeaw.ac.at. See also the Special Semester webpagewww.ricam.oeaw.ac.at/specsem/specsem2011/for additional information.

Figure 1: 4th floor of Science Park Building 2.

Figure 2: Campus plan

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Figure 3: Overview map

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Program

Monday, November 21st

00:00 - 09:40 Registration 09:40 - 09:50 Opening

09:50 - 10:40 Ralf Hiptmair (ETH Z¨urich)

“Convergence of truncated T-matrix approximation”

10:40 - 11:10 Coffee Break

11:10 - 12:00 Timo Betcke (University College London)

“Modulated plane wave methods for Helmholtz problems in heterogeneous media”

12:00 - 14:00 Lunch Break

14:00 - 14:50 Susan Minkoff (University of Maryland, Baltimore County)

“Two scale wave equation modeling”

14:50 - 15:20 Coffee Break

15:20 - 16:10 Paul Childs (Schlumberger Cambridge Research)

“Numerics of waveform inversion for seismic data”

16:10 - 17:00 Chris Stolk (University of Amsterdam)

“Seismic inverse scattering by reverse time migration”

17:15 Poster Session & Welcome Reception

Tuesday, November 22nd

09:00 - 09:50 Martin Gander (University of Geneva)

“Are absorbing boundary conditions and perfectly matched layers really so differ- ent?”

09:50 - 10:40 Lothar Nannen (Vienna University of Technology)

“Hardy space infinite elements for exterior Maxwell problems”

10:40 - 11:10 Coffee Break

11:10 - 12:00 Habib Ammari ( ´Ecole Normale Sup´erieure)

“Electromagnetic invisibility and super-resolution”

11:30 - 14:00 Lunch Break

14:00 - 14:50 Marcus Grote (University of Basel)

“Discontinuous Galerkin methods and local time stepping for transient wave prop- agation”

14:50 - 15:20 Coffee Break

15:20 - 16:10 Joachim Sch¨oberl (Vienna University of Technology)

“Hybrid discontinuous Galerkin finite element methods for the Helmholtz equation”

16:10 - 17:00 Break

17:15 - 18:00 Radon colloquiumby Martin Gander (University of Geneva)

“From Euler, Schwarz, Ritz and Galerkin to Modern Computing”

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Wednesday, November 23rd

09:00 - 09:50 Ronny Ramlau (University of Linz)

“Inverse problems in adaptive optics”

09:50 - 10:40 Fr´ed´eric Nataf (Universit´e Paris 6, Laboratoire J.L. Lions)

“Time reversed absorbing condition in the partial aperture case”

10:40 - 11:10 Coffee Break

11:10 - 12:00 Olaf Steinbach (Graz University of Technology)

“Boundary element methods for acoustic and electromagnetic scattering problems”

free afternoon (please feel free to make use of the seminar room and the Special Semester offices)

Thursday, November 24th

09:00 - 09:50 Roland Potthast (Deutscher Wetterdienst & University of Reading)

“Approaches to dynamical inverse scattering problems”

09:50 - 10:40 Thanh Nguyen (RICAM)

“Inverse obstacle scattering problems using multifrequency measurements”

10:40 - 11:10 Coffee Break

11:10 - 12:00 Guanghui Hu (WIAS Berlin)

“Direct and inverse scattering of elastic waves by diffraction gratings”

12:00 - 14:00 Lunch Break

14:00 - 14:50 Jari Toivanen (University of Jyv¨askyl¨a)

“Domain decomposition and multigrid preconditioners for the Helmholtz equation in layered and heterogeneous media”

14:50 - 15:20 Coffee Break

15:20 - 16:10 Radek Tezaur (Stanford University)

“The discontinuous enrichment method and its domain decomposition solver for the Helmholtz equation”

16:10 - 17:00 Ira Livshits (Ball State University)

“Algebraic multigrid algorithm for solving Helmholtz equations with large wave numbers”

19:00 Conference dinner

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Friday, November 25th

09:00 - 09:50 Manfred Kaltenbacher (University of Klagenfurt)

“Spectral finite elements for a mixed formulation in computational acoustics taking flow effects into acount”

09:50 - 10:40 Euan Spence (University of Bath)

“Stability and conditioning of boundary integral methods for high frequency scat- tering”

10:40 - 11:10 Coffee Break

11:10 - 12:00 Simon Chandler-Wilde (University of Reading)

“Numerical-Asymptotic Integral Equation Methods for High Frequency Scattering”

12:00 Closing

Posters

The poster session will take place on the 4th floor of the 2nd Science Park Building. It will start at 5:15 pmonMonday, November 21st.

Carlos Borges (Worcester Polytechnic Institute)

“Numerical solution of the high frequency 2D direct scattering problem for convex objects using non- uniform B-splines”

Victor A. Kovtunenko (University of Graz)

“Variational methods for the identification of objects”

Marie Kray (Universit´e Paris 6)

“Time reversed absorbing conditions: discrimination between one single inclusion and two close inclusions in a non-homogeneous medium”

Jens Markus Melenk (Vienna University of Technology)

“Wavenumber-explicit convergence analysis for the Helmholtz equation: hp-FEM andhp-BEM”

Andrea Moiola (ETH Z¨urich)

“Trefftz-discontinuous Galerkin methods for time-harmonic Maxwell’s equations”

Imbo Sim (University of Klagenfurt)

“Stable absorbing layer for convective wave propagation”

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Abstracts

“Electromagnetic invisibility and super-resolution”

Habib Ammari

Ecole Normale Sup´´ erieure

Abstract

The aim of this talk is threefold: - to give a mathematical justification of cloaking due to anomalous localized resonance; - to provide an original method for enhancing near cloaking; - to achieve resolved inclusion imaging.

“Modulated plane wave methods for Helmholtz problems in heterogeneous media”

Timo Betcke

Department of Mathematics University College London

Gower Street, London, WC1E 6BT, UK

Abstract

A major challenge in seismic imaging is full waveform inversion in the frequency domain. If an acoustic model is assumed the underlying problem formulation is a Helmholtz equation of the form

−∆u− ω

c(x) 2

u=f,

where c(x) is the varying speed of sound in the heterogeneous medium. Typically, in seismic ap- plications the solution u has many wavelengths across the computational domain, leading to very large linear systems after discretisation with standard finite element methods. Much progress has been achieved in recent years by the development of better preconditioners for the iterative solution of these linear systems. But the fundamental problem of requiring many degrees of freedom per wavelength for the discretisation remains.

For problems in homogeneous media, that iscis constant, plane wave finite element methods have gained significant attention. The idea is that instead of polynomials on each element we use a linear combination of plane waves of the form eiωcdj·x, where the dj are direction vectors of unit length.

These basis functions already oscillate with the right wavelength, leading to a significant reduction in the required number of unknowns. However, higher-order convergence is only achieved for problems with constant or piecewise constant media.

In this talk we discuss the use of modulated plane waves of the formp(x)eiωc¯dj·xfor problems in heterogeneous media, where pis a polynomial of low degree (typically 2 or 3) and ¯c is a constant approximation to the speed of sound in an element. The idea is that high-order convergence in a varying medium is recovered due to the polynomial modulation of the plane waves. Wave directions are chosen based on information from raytracing or other fast solvers for the eikonal equation. This approach is related to the Amplitude FEM originally proposed by Giladi and Keller in 2001. However, for the assembly of the systems we will use a discontinuous Galerkin method, which allows a simple way of incorporating multiple phase information in one element. We will discuss the dependence of the element sizes on the wavelenth and the accuracy of the phase information, and present several examples that demonstrate the properties of modulated plane wave methods for heterogeneous media problems.

“Numerical-asymptotic integral equation methods for high frequency scattering”

Simon Chandler-Wilde

Department of Mathematics and Statistics University of Reading

Reading RG6 6AX, UK

Abstract

Conventional discretisation methods for wave propagation and scattering (finite elements, finite difference, boundary element, ...) have costs which increase rapidly as the frequency increases because of the need for large numbers of degrees of freedom to resolve the oscillatory solution. In particular, this is an issue in boundary element calculations for time harmonic problems, where in 3D the degrees of freedom need to increase in proportion tok2, wherekis the wave number, in order to maintain a fixed number of degrees of freedom per wavelength, required to maintain accuracy. High frequency asymptotic methods, in the other hand, based on ray tracing/solving eikonal/transport equations,

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have a cost which is fixed ask increases, but have unacceptably low accuracy for many problems, except at very highk.

In this talk, we overview progress in developing numerical methods for high frequency problems which try to combine standard numerical and asymptotic approaches. In particular, we focus on boundary integral equation based methods, describing progress in generating numerical schemes which deliver an arbitrarily high requested accuracy with a number of degrees of freedom which provably (through numerical analysis theorems and numerical experiments) needs only grow logarithmically askincreases.

The methodology is to obtain knowledge of the phase structure of the solution, by rather elemen- tary high frequency ray analysis, and to build this phase structure into the basis functions used to approximate the solution. This idea, in a simple form, dates back at least to [1], but the methodology has seen a wealth of new ideas in the last 5-10 years, see [1–6] and the references therein, which we review.

[1] T. Abboud, J.C. N´ed´elec, B. Zhou, M´ethode des ´equations int´egrales pour les hautes fr´equencies, C.R. Acad. Sci. Paris.,318S´erie I (1994), 165-170.

[2] O.P. Bruno, F. Reitich, High Order Methods for High-Frequency Scattering Applications, In:

Modeling and Computations in Electromagnetics, H. Ammari (Ed.), Springer, 2007, pp. 129–164.

[3] S.N. Chandler-Wilde, I.G. Graham, Boundary Integral Methods in High Frequency Scattering, In:

Highly Oscillatory Problems, B. Engquist, T. Fokas, E. Hairer, A. Iserles (Eds.), Cambridge Univer- sity Press, 2009, pp. 154–193.

[4] M. Ganesh, S.C. Hawkins, A Fully Discrete Galerkin Method for High Frequency Exterior Acous- tic Scattering in Three Dimensions,J. Comp. Phys. 230, (2011), 104–125.

[5] A. Spence, S.N. Chandler-Wilde, I.G. Graham, V.P. Smyshlyaev, A New Frequency-Uniform Co- ercive Boundary Integral Equation for Acoustic Scattering,Comm. Pure Appl. Math.,64, (2011), 1384-1415.

[6] S.N. Chandler-Wilde, I.G. Graham, S. Langdon, E. Spence, Numerical-asymptotic boundary in- tegral methods in high frequency acoustic scattering, to appear inActa Numerica.

“Numerics of waveform inversion for seismic data”

Paul Childs

Schlumberger Cambridge Research

Abstract

Depth imaging and inversion of seismic data is becoming commonplace within the seismic explo- ration industry. However, the inversion procedures used today often fail to find the global minimum, and careful multiscale data processing must be included in the workflow. Although developed by Tarantola et al in the 1980s, there are still numerous mathematical challenges remaining relating to non-uniqueness, size of the null space of the inversion operator, and the sheer computational size of industrial datasets. In this talk, we will review some approaches to improving the robustness of the full waveform inversion method, which today is regarded as a highly interpretive procedure.

Because the PDE-constrained inversion procedure used in industry can be very expensive due to the large number of PDE solves required, we will review and address some of the numerical challenges in this area. The talk will serve as an introduction to the subject and will concentrate mainly on computational developments. Illustrations will be given with industrial examples from full waveform inversion of seismic data.

“Are absorbing boundary conditions and perfectly matched layers really so different?”

Martin J. Gander Section of Mathematics University of Geneva

2-4 Rue du Li`evre, CH–1211 Gen`eve 4

Abstract

In order to truncate infinite computational domains, there are two major competitors: absorbing boundary conditions and perfectly matched layers. The former use approximations of the Dirichlet to Neumann map, and the latter adds a layer outside the truncated domain, in which a modified equation is solved, which absorbs outgoing parts of the solution.

Using the concept of the pole condition, we show for a model problem that these two seem- ingly very different techniques are in fact related. For the particular case of an absorbing boundary condition given by a Pade approximation of the Dirichlet to Neumann map, we show that a nat- ural implementation leads to a layer structure corresponding to a perfectly matched layer with an exponentially scaled outer problem.

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“From Euler, Schwarz, Ritz, and Galerkin to modern computing”

Martin J. Gander

Section of Mathematics, University of Geneva, 2-4 Rue du Li`evre, CH–1211 Gen`eve 4 Abstract

The origins of modern computing are dispersed over centuries, often in the work of pure mathe- maticians, who invented methods in order to understand mathematical objects and prove theorems.

A typical example is the famous Schwarz method for parallel computing, whose origins lie in a prob- lem in Riemann’s audacious proof of the Riemann mapping theorem. Another example is the finite element method, which has its origins in the variational calculus of Euler-Lagrange and in the thesis of Walther Ritz, who died just over 100 years ago at the age of 31 from tuberculosis. We will see in this talk that the path leading to modern computational methods and theory was a long struggle over three centuries requiring the efforts of many great mathematicians.

“Discontinuous Galerkin methods and local time stepping for transient wave propagation”

Marcus J. Grote Mathematisches Institut

Rheinsprung 21, CH–4051 Basel

Abstract

The accurate and reliable simulation of transient wave phenomena is of fundamental importance in a wide range of engineering applications such as fiber optics, wireless communication, seismic imaging, and non-invasive testing. Finite element methods are probably the most flexible approach for computational wave propagation in heterogeneous media or complex geometry. When combined with explicit time integrators, standard conforming finite element methods lead to efficient and highly parallel methods, if mass-lumping techniques are used. Alternatively, discontinuous Galerkin methods easily accommodate irregular non-matching grids and hp-refinement, while they inherently lead to block-diagonal mass matrices. Hence, they also lead to efficient and highly parallel methods when combined with explicit time-stepping schemes. Nonetheless, both suffer from the severe stability (CFL) condition imposed on the time step by the smallest elements in the mesh. To circumvent that CFL condition, we propose high-order explicit local time-stepping schemes, which allow smaller time steps precisely where the smallest elements in the mesh are located.

This is joint work with J. Diaz (INRIA), T. Mitkova (Basel) and D. Schoetzau (UBC).

“Direct and inverse scattering of elastic waves by diffraction gratings”

Guanghui Hu WIAS Berlin Mohrenstr. 39 D–10117 Berlin

Abstract

This talk is concerned with the scattering of a time-harmonic plane elastic wave by an unbounded periodic structure. Such structures are also called diffraction gratings and have many important applications in diffractive optics, radar imaging, non-destructive testing, etc.

The talk is arranged into three sections. (i) Variational approach to scattering of plane elastic waves by an impenetrable two-dimensional periodic structure. Using a variational formulation in a bounded periodic cell involving a non-local boundary operator, uniqueness and existence of quasi- periodic solutions are presented under the first (Dirichlet), second (Neumann), third and fourth kind boundary conditions. (ii) Uniqueness for the inverse problem in determining a polygonal diffraction grating from near-field measurements under the third or fourth kind boundary conditions. We deter- mine and classify all the unidentifiable grating profiles corresponding to one given incident pressure or shear wave, relying on the reflection principle of the Navier equation. (iii) An optimization method for the inverse problem in recovering an unknown grating profile from scattered elastic waves measured above the structure. Based on the Kirsch-Kress optimization scheme, we apply a two-step algorithm to both smooth and piecewise linear gratings for the Dirichlet boundary value problem of the Navier equation. Numerical reconstructions from exact and noisy data will be shown.

Some results for bi-periodic structures will also be mentioned. This is joint work with Dr. J.

Elschner under the DFG Project: ”Direct and inverse scattering problems for elastic waves”

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“Convergence of truncated T-matrix approximation”

Ralf Hiptmair

Seminar for Applied Mathematics ETH Z¨urich

Abstract

The T-matrix encodes the scattering properties of an obstacle independently of the incident and receiver directions. It boils down to a basis representation of the operator mapping incident fields to the far field pattern with respect to (spherical) harmonics. Discretization can be achieved by truncating the T-matrix, which amounts to using a finite number of basis functions. The truncated T-matrix can be computed by solving several large forward scattering problems during a preparatory offline stage. Then, the scattering response to an incident field of any direction or source position can be computed very fast (online stage).

My presentation is concerned with the a priori analysis of the effect of truncating the T-matrix for acoustic scattering. Detailed error estimates and predictions of the rates of exponential convergence are derived for both point-source and plane-wave incident waves. Errors in solving the forward problem are also taken into account.

This is joint work with M. Ganesh (Colorado School of Mines) and S.C. Hawkins (Macquarie University, Sydney).

References.

M. Ganesh, S. Hawkins, and R. Hiptmair, Convergence analysis with parameter estimates for a reduced basis acoustic scattering T-matrix method, Report 2011-04, SAM, ETH Z¨urich, Z¨urich, Switzerland, 2011. To appear in IMA J. Numer. Anal.

“Spectral finite elements for a mixed formulation in computational acoustics taking flow effects into ac- count”

Manfred Kaltenbacher Applied Mechatronics University of Klagenfurt Austria

Abstract

Formulations which include effects of a mean flow on the acoustic wave propagation are mostly based on a splitting of the unknowns velocity~u, pressurepand densityρinto mean (denoted by an overline) and fluctuating (denoted by the superscript a) parts. Such an ansatz results, e.g., in the acoustic perturbation equations as used in computational aeroacoustics [1]

∂pa

∂t +c2ρ∇ ·~ua+~u· ∇pa=qc; ρ∂~ua

∂t +ρ(~u· ∇)~ua+∇pa=~qm

withcthe speed of sound andqc,~qmacoustic source terms. Its variational formulation is then given as follows (for simplification we use homogeneous Dirichlet boundary conditions): Find (~u , p0)∈V×W such that

∂t Z

paϕ,dx−c2ρ Z

~

ua· ∇ϕdx+ Z

~u· ∇padx= Z

qcϕdx (1)

ρ∂

∂t Z

~

va·ψ~dx+ Z

∇pa·ψ~dx+ρ Z

(~u· ∇)~uadx= Z

~

qm·ψ~dx (2)

for all (ψ , ϕ)~ ∈V ×W.

Following [2] we apply a mixed formulation and use spectral elements with the following discrete spaces

Wh =

ϕh∈H01

ϕh|Kj◦Fj∈ PN , Vh = n

φ~h∈[L2]d

|Jj|Jj−1φ|~Kj◦Fj∈[PN]do .

In order to stabilize our formulation, we use similar techniques as for DG-approaches and reformulate the third term in (2) as a special flux term (upwinding) and furthermore add a jump-term in the acoustic particle velocity~ua along common element interfaces.

This topic is a joint work with A. H¨uppe (University of Klagenfurt), G. Cohen and S. Imperial (IN- RIA, Paris) and B. Wohlmuth (TU Munich).

[1] R. Ewert and W. Schr¨oder. Acoustic perturbation equations based on flow decomposition via source filtering. Journal of Comp. Phys., 188:365398, 2003.

[2] G. Cohen and S. Fauqueux. Mixed finite elements with mass-lumping for the transient wave equation. Journal of Comp. Acoustics, 8:171188, 2000.

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“Algebraic multigrid algorithm for solving Helmholtz equations with large wave numbers”

Ira Livshits

Department of Mathematical Sciences Ball State University

USA

Abstract

In this talk we introduce a new adaptive algebraic multigrid approach for solving indefinite Helmholtz equations. The proposed approach is an interplay between the idea of multiple smooth representations of oscillatory kernel components of the Helmholtz operator on coarse grids, as in the geometric multigrid wave-ray algorithm, and a flexibility of a more recent least-squares Bootstrap AMG approach, which has demonstrated its potential in different types of applications. We present preliminary results in two dimensions and discuss future directions.

“Two Scale Wave Equation Modeling”

Susan E. Minkoff

Department of Mathematics and Statistics University of Maryland, Baltimore County 1000 Hilltop Circle

Baltimore, MD 21250, USA Email: [email protected]

Abstract

Imaging the Earth’s subsurface requires determination of the “important information” inherent in data that ranges over multiple scales. While numerical upscaling is a common approach for speeding up solution of the fluid flow equations, simulation of waves through the same Earth is generally accomplished using single scale finite differences. We describe a two-scale finite-element based wave simulator and the associated adjoint problem. Operator-based upscaling decomposes the solution into coarse and subgrid components. The subgrid solution lives at the fine nodes inside each coarse cell. Fine-scale solution information is incorporated in the coarse solution. After adapting operator- based upscaling to the acoustic and elastic wave equations in 2 and 3D, we give a matrix analysis of this two-stage process that produces the first explanation of the underlying physical equations being solved by this technique. We see that the coarse grid solution involves solving a matrix problem with entries that are averages of the original parameter field on coarse grid boundaries. Calculation of the adjoint for the upscaled wave equation simulator is straight-forward if differentiation of the continuous pde model is accomplished before discretization. The result is that the adjoint problem can be solved by the same upscaling method as the standard acoustic wave equation.

“Hardy space infinite elements for exterior Maxwell problems”

Lothar Nannen

Institute for Analysis and Scientific Computing Vienna University of Technology

Austria

Abstract

In this talk we present an infinite element method for solving electromagnetic scattering and resonance problems posed on unbounded domains. As our motivation is to solve Maxwell’s equations we take care that these infinite elements fit into the discrete de Rham diagram, i.e. they span discrete spaces, which together with the exterior derivative form an exact sequence.

The theoretical framework of the method is the so called pole condition, which characterizes radiating solutions via the poles or singularities of the Laplace transformed solutions: The Laplace transform in radial direction of an outgoing wave belongs to a certain Hardy space of holomorphic functions, while the Laplace transform of an incoming wave does not. Hence, the Hardy space infinite elements are constructed using tensor products of Hardy space basis functions with standard finite element surface basis functions.

Numerical tests indicate super-algebraic convergence in the number of additional unknowns per degree of freedom on the coupling boundary.

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“Time reversed absorbing condition in the partial aperture case”

Fr´ed´eric Nataf

Laboratoire J.L. Lions and CNRS UMR7598 University Pierre & Marie Curie

Paris, France

Abstract

The time-reversed absorbing conditions (TRAC) method introduced inTime Reversed Absorbing Condition: Application to Inverse Problems, Assous, Kray and Nataf (Inverse Problems, 2011) enables one to “recreate the past” without knowing the source which has emitted the signals that are back- propagated. It has been applied to inverse problems for the reduction of the computational domain size and for the determination, from boundary measurements, of the location and volume of an unknown inclusion. The method does not rely on anya prioriknowledge of the physical properties of the inclusion. We present the extension of theTRACmethod to the partial aperture configuration and to discrete receivers with various spacing. In particular the TRACmethod is applied to the differentiation between a single inclusion and a two close inclusion case. Subwavelength resolution can be achieved even with more than 20% noise in the data.

“Inverse obstacle scattering problems using multifrequency measurements”

Trung Th`anh Nguyen RICAM

Altenbergerstrasse 69, A-4040 Linz

Abstract

In this talk, we consider the problem of reconstructing the shapes of sound-soft acoustic obstacles using far field measurements associated with only one or a few incident directions but at multiple frequencies. The motivation for using multifrequency data is explained as follows. On the one hand, we know that the reconstruction problem is uniquely solvable at low frequencies, but its stability is poor (only of log-type). That means, at low frequencies, it is difficult to reconstruct small details of the obstacle. On the other hand, at high frequencies, this inverse problem may not be uniquely solvable but it is more stable. To take the advantages of both low and high frequencies, we use the fol- lowing approach. We first reconstruct a rough approximation of the obstacles at the lowest frequency using the least-squares approach. This reconstruction is then refined by using recursive optimization methods at higher frequencies. This approach enables us to obtain an accurate reconstruction of the parts of the obstacles’ boundary illuminated by the incident waves without requiring a good initial guess. The analysis of this approach is divided into three steps.

In the first step, we derive a quantitative estimate of the set of local convexity of the objective functional at a fixed frequency. Our analysis shows that, the size of this set is inversely proportional to the used frequency. As a consequence, if the obstacles are expected to be contained in a known domain, the lowest frequency should be chosen small enough so that the set of convexity of the objective functional contains this domain and any shape in this domain can be used as an initial guess.

The appropriate choice of the lowest frequency avoids the need of a good initial guess to obtain an approximation of the true shapes. However, due to the lack of good stability at low frequencies, we can only expect a rough approximation. To enhance the accuracy, our idea is then to use this rough approximation as an initial guess for minimizing the objective functional at higher frequencies.

For this purpose, we make use of recursive optimization methods. The second step is devoted to the convergence of the recursive optimization algorithms.

To analyze the reconstruction accuracy, in the third step, we discuss the stability at high frequen- cies. We justify a conditional asymptotic Lipschitz stability estimate of the so-called support function on the parts of the obstacles’ boundary illuminated by the incident wave. This result explains why we can reconstruct small details of the illuminated parts at high frequencies.

The results presented are joint work with Mourad Sini (RICAM).

“Approaches to dynamical inverse scattering problems”

Roland Potthast

Deutscher Wetterdienst, University of Reading, University of G¨ottingen Abstract

We discuss several different approaches to dynamics inverse scattering problems. Dynamical problems are widely spread in nature. Dynamics can refer to the dynamics to the scattered field, i.e. when we have time-dependent fields. In this case several new methods have been developed

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over the past years and we will describe dynamical approaches like the time-domain probe method and orthogonality sampling. When the scatterers under consideration are time-dependent, we need additional techniques from the area of data assimilation. We will provide a survey for the case where scale separation can be employed for the time-scales of the scatterer and the waves.

“Inverse problems in adaptive optics”

Ronny Ramlau

Institute for Industrial Mathematics Johannes Kepler University Linz Austria

Abstract

Most of the large earthbound astronomical telescopes use Adaptive Optics technology (AO) in or- der to enhance the image quality. The degradation of the measured images is caused by atmospheric turbulences. The correction is achieved by the use of deformable mirrors, where the deformation is obtained from the measurements of the light of a bright natural star or artificial stars (laser guide stars). We consider Inverse Problems that arise form Single Conjugate Adaptive Optics (SCAO) and Multi Conjugate Adaptive Optics (MCAO). In order to measure the incoming wavefront, different types of sensors are used. We consider the so called Shack - Hartmann sensor which measures an average of the gradient of the wavefront. For SCAO, the Inverse Problem is now the reconstruc- tion of the wavefront from the (noisy) sensor measurements. Besides the reconstruction quality the reconstruction time is most important, as the reconstructions have to be carried out in real time.

The presented reconstruction algorithms for SCAO are also an ingredient for the computation of the mirror deformation for MCAO, which is based on a tomography of the atmosphere. The analytical results are illustrated by numerical experiments.

“Hybrid discontinuous Galerkin finite element methods for the Helmholtz equation”

Joachim Sch¨oberl

Institute for Analysis and Scientific Computing Vienna University of Technology

Abstract

In this paper we present Hybrid Discontinuous Galerkin (HDG) Methods to discretize the Helmholtz equation. In constrast to elliptic equations, we introduce two hybrid variables, one for the Dirichlet and one for Neumann data. Thanks to this choice, the element equations can be formulated using impedance traces.

We present computational results for iterative solvers with BDDC–type domain decomposition preconditioners.

This talk contains joint work with A. Pechstein, P. Monk, M. Huber, A. Hannukainen, and G.

Kitzler.

“Stability and conditioning of boundary integral methods for high frequency scattering”

Euan A. Spence

Department of Mathematical Sciences University of Bath

Bath, BA2 7AY, UK

Abstract

Boundary integral equations are a classical tool for solving acoustic scattering problems modelled by the Helmholtz equation.

This talk will discuss the following two questions:

1. How do the condition numbers of the integral operators depend on the wavenumber k (in particular forklarge)?

2. Can one give ak-explicit error analysis, (a) for methods using conventional piecewise polynomial basis functions (which require the number of degrees of freedom to grow as k increases to preserve accuracy), and (b) for methods using novelk-dependent basis functions (which have the potential to give almost uniform accuracy independent of k– see the next talk by Simon Chandler-Wilde)?

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Both these questions are interesting because the standard analysis of boundary integral equations for the Helmholtz equation doesnot yieldk-explicit results.

This talk will give an overview of some recent developments in answering these questions and will discuss joint work with Timo Betcke (University College London), Simon Chandler-Wilde (Reading), Ivan Graham (Bath), and Valery Smyshlyaev (University College London), as well as independent work by Markus Melenk (Vienna).

“Boundary element methods for acoustic and electromagnetic scattering problems”

Olaf Steinbach

Institut f¨ur Numerische Mathematik TU Graz

Steyrergasse 30 8010 Graz Austria

Abstract

For the numerical solution of acoustic and electromagnetic scattering problems we discuss sta- bilized boundary integral formulations and related boundary element methods. For problems with piecewise local wave numbers we apply a tearing and interconnecting domain decomposition approach which is based on modified Robin interface conditions to ensure unique solvability.

The talk is based on joint work with Sarah Engleder and Markus Windisch.

“Seismic inverse scattering by reverse time migration”

Chris Stolk

University of Amsterdam

Korteweg-de Vries Institute for Mathematics Science Park 904

1098 XH, Amsterdam The Netherlands

Abstract

We will consider the linearized inverse scattering problem from seismic imaging. While the first reverse time migration algorithms were developed some thirty years ago, they have only recently be- come popular for practical applications. We will analyze a modification of the reverse time migration algorithm that turns it into a method for linearized inversion, in the sense of a parametrix. This is proven using tools from microlocal analysis. We will also discuss the limitations of the method and show some numerical results.

“The discontinuous enrichment method and its domain decomposition solver for the Helmholtz equation”

Radek Tezaur Stanford University

Aeronautics and Astronautics 496 Lomita Mall

Stanford, CA 94305-4035, USA

Abstract

The Discontinuous Enrichment Method (DEM) is a discretization method designed for an effi- cient solution of multi-scale problems. It is based on a hybrid variational formulation with Lagrange multipliers. In addition to an optional polynomial field, it employs free-space solutions of the gov- erning differential equation for approximating large gradients or highly oscillatory components of the solution. It relies on Lagrange multipliers to enforce a weak form of the inter-element continuity of the solution. Recent developments and applications of DEM are discussed. These include the solution of acoustic scattering and structural dynamics problems in the medium frequency regime, and advection-diffusion problems with high Peclet numbers. Comparisons to other method are shown and variable coefficient ideas are discussed.

Then, a nonoverlapping domain decomposition method is presented for the solution of Helmholtz/structural dynamics problems discretized by DEM with and without the optional polynomial field. In this do-

main decomposition method, the primal subdomain degrees of freedom are eliminated by local static condensations to obtain an algebraic system of equations formulated in terms of the interface Lagrange multipliers only. As in the FETI-H and FETI-DPH domain decomposition methods for continuous

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Galerkin discretizations, this system of Lagrange multipliers is iteratively solved by a Krylov method equipped with both a local preconditioner based on subdomain data, and a global one using a coarse space. Numerical experiments performed for two- and three-dimensional acoustic problems demon- strate the scalability of the method with respect to the size of the global problem and the number of subdomains.

The principal co-authors of this talk are Charbel Farhat and Jari Toivanen. Other co-authors will be acknowledged during the talk.

“Domain decomposition and multigrid preconditioners for the Helmholtz equation in layered and hetero- geneous media”

Jari Toivanen

Department of Mathematical Information Technology, University of Jyv¨askyl¨a FI-40100 Jyv¨askyl¨a

Finland

Abstract

We consider the efficient numerical solution of time-harmonic acoustic scattering problems.

For layered media, we propose a domain decomposition preconditioner which is based on domain embedding subdomain preconditioners and a fast direct solver. The iterations can be carried out on small subspaces associated to the interfaces between layers/subdomains. The numerical experiments demonstrate the capability to solve very large scale two-dimensional and three-dimensional problems with up to billions of unknowns. The number of iterations seems to behave roughly logarithmically with respect to the frequency.

For heterogeneous media, we consider a multigrid preconditioner which is based on an algebraic multigrid method and a physically damped operator. Numerical experiments show this approach to lead to efficient solution procedure for low and medium frequency problems. The number of iterations grows roughly linearly with respect to the frequency.

This is joint work with Tuomas Airaksinen (University of Jyv¨askyl¨a) and Kazufumi Ito (North Carolina State University, Raleigh, NC).

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Abstracts for Posters

“Numerical solution of the high frequency 2D direct scattering problem for convex objects using non- uniform B-splines”

Carlos Borges

Worcester Polytechnic Institute 100 Institute Rd

Worcester, MA USA

Abstract

We present a method for the numerical solution of the acoustic two dimensional direct scattering problem for non penetrable convex scatterers for high frequencies. The method takes advantage of the fact that, for the integral formulation considered for the problem, the solution to the integral equation is a physical quantity and thereby can be written as the product of a known highly oscillatory function and a unknown slow oscillatory function. We solve the problem for the slow oscillatory function using a collocation method and we approximate the solution by a linear combination of non-uniform B- splines, where the density of control points corresponding will be higher where more resolution is needed. To perform the integrations, we use ideas of the method of stationary phase, techniques of integration of high oscillatory functions and singular kernels. Numerical results of the proposed method are presented.

“Variational methods for the identification of objects”

Victor A. Kovtunenko

Institute for Mathematics and Scientific Computing KF-University of Graz

Heinrichstr.36, A-8010 Graz

Lavrent’ev Institute of Hydrodynamics 630090 Novosibirsk, Russia,

e-mail: [email protected]

Abstract

Topology optimization problems for identification and reconstruction of small geometric objects of arbitrary shapes and unknown boundary conditions are addressed. By arbitrary shapes we mean the most general, i.e., the least restrictive, geometric assumptions which are commonly used in variational formulations. We refer to singular geometric objects such as thin obstacles, cracks, multi-dimensional junks, and alike. For unknown boundary conditions we consider Robin type boundary conditions described by unknown parameters, which are discrete or distributed. Thus, the boundary conditions are determined a-posteriori, i.e., after solving the optimization problem.

From a mathematical point of view, identification of unknown objects is an inverse problem, which belongs to the field of shape and topology optimization and parameter estimation. Our approach to the identification problems is based on variational methods supported with generalized methods of singular perturbations. The key ingredients of the optimization approach are as follows. Firstly, we consider the identification problem as a state-constrained optimization problem and apply proper duality principles providing us with optimality conditions in the weak setting, thus accounting for the features above. Second, we employ asymptotic methods of singular perturbation theory to obtain approximate, geometry-dependent models. While the the first-order terms are somewhat known in the literature as the topological derivative, we get the second-order terms to determine the boundary conditions. Third, from multi-parametric optimization we get semi-analytical solution strategies.

We consider an example of inverse scattering problem. Some numerical aspects of the topology optimization will be discussed.

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“Time reversed absorbing conditions:

discrimination between one single inclusion and two close inclusions in a non-homogeneous medium”

Marie Kray

Jacques-Louis Lions Laboratory, Pierre et Marie Curie University Paris, France

[email protected]

Abstract

We introduce the time-reversed absorbing conditions (TRAC) in time-reversal methods. They enable one to “recreate the past” without knowing the source which has emitted the signals that are back-propagated. We present two applications in inverse problems: the reduction of the size of the computational domain and the determination, from boundary measurements, of the location and volume of an unknown inclusion. The method does not rely on any a priori knowledge of the physical properties of the inclusion. Numerical tests with the wave equation illustrate the efficiency of the method. Moreover theTRACmethod is fairly insensitive with respect to the magnitude of noise on the recorded data.

In order to be closer to realistic cases as in geophysics, mine detection, medical imaging, ..., we consider non homogeneous media (random or layered media). The application we propose is to in- vestigate the ability of theTRACmethod to discriminate a unique inclusion from two distinct close inclusions in a non-homogeneous medium. This test is inspired by a more realistic setting. Our intent is to detect one or two iron or plastic mines in a background medium that can be random or layered.

The physical equation we use is a scalar wave equation derived from the Maxwell equations.

Joint work with Franck Assous (Bar-Ilan University & Ariel University Center, Israel, [email protected]) and Fr´ed´eric Nataf (Universit´e Paris 6,[email protected]) Publications:

• F. Assous, M. Kray, F. Nataf,Time Reversed Absorbing Condition in the Partial Aperture Case, http://hal.archives-ouvertes.fr/hal-00581291/fr/

• F. Assous, M. Kray, F. Nataf, E. Turkel,Time Reversed Absorbing Condition: Application to inverse problem, Inverse Problems (2011), Vol. 27, no 6, pp 065003.

• F. Assous, M. Kray, F. Nataf, E. Turkel,Time Reversed Absorbing Conditions, Comptes Rendus Mathmatiques, Serie I (2010), Vol. 348, no 19-20, pp 1063-1067, doi:10.1016/j.crma.2010.09.014 .

“Wavenumber-explicit convergence analysis for the Helmholtz equation: hp-FEM and hp-BEM”

J.M. Melenk

Vienna University of Technology

Abstract

We consider boundary value problems for the Helmholtz equation at large wave numbersk. In order to understand how the wave numberkaffects the convergence properties of discretizations of such problems, we develop a regularity theory for the Helmholtz equation that is explicit ink. At the heart of our analysis is the decomposition of solutions into two components: the first component is an analytic, but highly oscillatory function and the second one has finite regularity but features wavenumber-independent bounds.

This understanding of the solution structure opens the door to the analysis of discretizations of the Helmholtz equation that are explicit in their dependence on the wavenumberk. As a first example, we show for a conforming high order finite element method that quasi-optimality is guaranteed if (a) the approximation orderpis selected asp=O(logk) and (b) the mesh sizehis such that kh/p is small. As a second example, we consider combined field boundary integral equation arising in acoustic scattering. Also for this example, the same scale resolution conditions as in the high order finite element case suffice to ensure quasi-optimality of the Galekrin discretization.

This work presented is joint work with Stefan Sauter (Z¨urich) and Maike L¨ohndorf (Vienna).

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“Trefftz-discontinuous Galerkin methods for time-harmonic Maxwell’s equations”

Andrea Moiola

Seminar for Applied Mathematics ETH Z¨urich

R¨amistrasse 101, CH–8092 Z¨urich [email protected]

Abstract

The propagation and the interaction of time-harmonic electric waves are described by the Maxwell equations ∇ ×(∇ ×E)−ω2E =0. The need of resolving oscillating solutions and the so-called pollution effect make their numerical discretization through standard finite element schemes extremely expensive for high frequencies. The Trefftz methods offer a way to tackle this problem: the trial and the test functions are solution of the PDE inside each element, thus they are oscillating functions and we can expect to be able to approximate the solution with smaller discrete spaces. The basis functions are often chosen as vector plane waves.

We formulate a class of Trefftz-discontinuous Galerkin (TDG) methods that includes the well- known ultra weak variational formulation (UWVF) of Cessenat and Despr´es, and we study the convergence of theirp-version (spectral version) for an impedance boundary value problem. Well- posedness and quasi-optimality in a mesh skeleton norm follow from the coercivity of the bilinear form defining the method on the Trefftz function space. A novel vector Rellich-type identity allows to prove new (wavenumber-independent) stability and regularity estimates for the considered boundary value problem in star-shaped polyhedral domains; these estimates, in turn, are used in a duality argument to prove error bounds in a mesh-independent norm.

The abstract convergence analysis is carried out for any Trefftz trial space. In the case of vector plane or spherical waves, concrete error bounds are obtained by using best approximation estimates for general Maxwell solutions that are proved from the analogous scalar (Helmholtz) results. The convergence is algebraic both in the dimension of the local trial space and in the meshwidth and all the bounds are explicit in the wavenumber.

This work is part of my PhD thesis, carried out under the supervision of Ralf Hiptmair (ETH Z¨urich) and Ilaria Perugia (Universit`a di Pavia).

“Stable absorbing layer for convective wave propagation”

Imbo Sim

University of Klagenfurt

Abstract

In a multi-model approachfor computational aeroacoustics (CAA) one often uses within the acous- tic source region some kind of acoustic perturbation equations and then in the ambient region the convective wave equation. Thereby, an efficient approach to model free field radiation conditions are required for the convective wave equation. The perfectly matched layer (PML) approach has proved a flexible and accurate method, which consists in surrounding the computational domain by an ab- sorbing layer. However, most PML formulations require wave equations stated in their standard second-order form to be reformulated as first-order hyperbolic systems, thereby introducing many additional unknowns. Here, we propose instead a simple stable PML formulation directly for the second-order convective wave equation both in two and in three space dimensions.

The work presented is joint work with Manfred Kaltenbacher (Klagenfurt).

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List of Participants

Akindeinde Saheed Ojo RICAM [email protected]

Alyaev Sergey University of Bergen [email protected]

Ammari Habib Ecole Normale Sup´´ erieure [email protected]

Arnold Thomas WIAS Berlin [email protected]

Betcke Timo University College London [email protected] Borges Carlos Worcester Polytechnic Institute [email protected]

Buck Marco Fraunhofer Institute for

Industrial Mathematics

[email protected] Buckwar Evelyn Johannes Kepler Universit¨at

Linz

[email protected]

Challa Durga Prasad RICAM [email protected]

Chandler-Wilde Simon University of Reading [email protected]

Childs Paul Schlumberger Cambridge

Research

[email protected] Engl Heinz RICAM & University of Vienna [email protected] Ernst Oliver TU Bergakademie Freiberg [email protected]

Freitag Melina University of Bath [email protected]

Gander Martin University of Geneva [email protected]

Georgiev Ivan RICAM [email protected]

Gessese Alelign University of Canterbury [email protected].

Graham Ivan University of Bath [email protected]

Grote Marcus University of Basel [email protected]

Heged¨us Gabor RICAM [email protected]

Helin Tapio RICAM [email protected]

Hiptmair Ralf ETH Z¨urich [email protected]

Hoang Viet Ha NTU, Singapore [email protected]

Hrtus Rostislav Institute of Geonics of the AS CR, v.v.i., Ostrava, CZ

[email protected]

Hu Guanghui WIAS Berlin [email protected]

Kaltenbacher Barbara University of Klagenfurt [email protected] Kaltenbacher Manfred University of Klagenfurt [email protected]

Kar Manas RICAM [email protected]

Karer Erwin RICAM [email protected]

Kollmann Markus Doctoral Program

Computational Mathematics, Johannes Kepler University

[email protected]

Kolmbauer Michael Institute of Computational Mathematics

[email protected] Kovtunenko Victor KF-University Graz; Lavrent’ev

Institute of Hydrodynamics

[email protected]

Kraus Johannes RICAM [email protected]

Kray Marie Laboratoire J-L Lions, UPMC [email protected]

Langer Ulrich University of Linz [email protected]

Leindl Mario Montanuniversit¨at Leoben, Institut f¨ur Mechanik

[email protected]

Livshits Irene Ball State University [email protected]

Melenk Jens Markus TU Wien [email protected]

Migliorati Giovanni Department of Mathematics, Politecnico di Milano

[email protected]

Minkoff Susan Dept. of Mathematics and

Statistics, University of Maryland Baltimore County

[email protected]

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Mitrovic Darko University of Bergen [email protected] Mohamed Menad University of Chlef

Moiola Andrea SAM - ETH Z¨urich [email protected]

Motamed Mohammad KAUST [email protected]

Nannen Lothar TU Vienna [email protected]

Nataf Fr´ed´eric Laboratoire J.L Lions and CNRS

[email protected]

Naumova Valeriya RICAM [email protected]

Nayak Sridhara Indian Institute of Technology Kharagpur

[email protected]

Nguyen Trung Thanh RICAM [email protected]

Nordbotten Jan Martin University of Bergen [email protected] Pechstein Clemens Institute of Computational

Mathematics, Johannes Kepler University Linz

[email protected]

Polydorides Nick Cyprus Institute [email protected]

Potthast Roland Deutscher Wetterdienst, Univ.

of Reading, Uni G¨ottingen

[email protected] Ramlau Ronny Industrial Mathematics

Institute, Kepler University Linz, Austria

[email protected]

Rieder Andreas Department of Mathematics, Karlsruhe Institute of Technology

[email protected]

Sarkis Marcus Mathematical Sciences Dept./Worcester Polytechnic Institute

[email protected]

Scheichl Robert University of Bath [email protected]

Schicho Josef RICAM [email protected]

Sch¨oberl Joachim Vienna UT [email protected]

Shanks Douglas University of Bath [email protected]

Sim Imbo University of Klagenfurt [email protected]

Sini Mourad RICAM [email protected]

Sokol Vojtˇech Institute of Geonics of the AS CR, v.v.i., Ostrava, CZ

[email protected]

Spence Euan University of Bath [email protected]

Spillane Nicole Laboratoire Jacques Louis Lions (Paris) - Michelin

[email protected]

Steinbach Olaf TU Graz [email protected]

Stolk Chris University of Amsterdam [email protected]

Talagrand Olivier Laboratoire de M´et´eorologie Dynamique, ´Ecole Normale Sup´erieure, Paris, France

[email protected]

Teckentrup Aretha University of Bath, Department of Mathematical Sciences

[email protected]

Tezaur Radek Stanford University [email protected]

Toivanen Jari Stanford University [email protected]

Tomar Satyendra RICAM [email protected]

Wachsmuth Daniel RICAM [email protected]

Willems J¨org RICAM [email protected]

Wohlmuth Barbara Technische Universit¨at M¨unchen

[email protected] Wolfmayr Monika Institute of Computational

Mathematics, University of Linz

[email protected]

Yang Huidong RICAM [email protected]

Zikatanov Ludmil The Pennsylvania State University

[email protected] Zulehner Walter Johannes Kepler University

Linz

[email protected]

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