Error analysis and fast solvers for high-frequency scattering problems

Size: px
Start display at page:

Download "Error analysis and fast solvers for high-frequency scattering problems"

Transcription

1 Error analysis and fast solvers for high-frequency scattering problems I.G. Graham (University of Bath) Woudschoten October 2014

2 High freq. problem for the Helmholtz equation Given an object Ω R d, with boundary Γ and exterior Ω, Incident plane wave, e.g. : u I (x) = exp(ikx â) Ω Ω â Γ Total wave u = u I + u S, where Scattered wave u S satisfies: plus boundary condition u S + k 2 u S = 0 in Ω (Mostly u I + u S = 0 on Γ) and radiation condition: us r ikus = o(r (d 1)/2 ) as r

3 Numerical-asymptotic methods Constant wavenumber k - asymptotic information (mostly BEM) Computing in time independent of frequency. Links to Daan and Simon s talks Geometry dependent methods

4 Numerical-asymptotic methods Constant wavenumber k - asymptotic information (mostly BEM) Computing in time independent of frequency. Links to Daan and Simon s talks Geometry dependent methods

5 Numerical-asymptotic methods Still a role for conventional BEM

6 Second Talk: Truncated problems u + k 2 u = 0 in Ω B R u = u I on Γ u n iku = 0 on B R B R Ω Γ Ω for large R Model cavity problem u + k 2 u = f in bounded domain Ω u iku n = g on Γ := Ω

7 Heterogeneity Seismic inversion problem: ( ) ωl 2 u u = f, c(x) ω = frequency solve for u with approximate c. Second talk: Conventional discretisation and fast solvers Key reference: Erlangga, Osterlee, Vuik, Link to Martin s talks and Domain Decomposition

8 Outline of my talks: Two problems on conventional methods. 1. When is the error in the h version BEM bounded independently of k? 2. Give an analysis of preconditioning methods for standard h version FEM Both have solutions which use high-frequency analysis.

9 References for the talks: I.G. Graham, M. Loehndorf, J.M. Melenk and E.A. Spence, When is the error in the h-bem for solving the Helmholtz equation bounded independently of k? To appear in BIT, M. J. Gander, I. G. Graham and E. A. Spence, How should one choose the shift for the shifted Laplacian to be a good preconditioner for the Helmholtz equation?, submitted I.G. Graham, E.A. Spence and E. Vainikko, Convergence of additive Schwarz methods for the Helmholtz equation with and without absorption, in preparation. S.N. Chandler-Wilde, I.G. Graham, S. Langdon and E.A. Spence, Boundary Integral Equation Methods for High Frequency Scattering Problems Acta Numerica 2012.

10 References for the talks: I.G. Graham, M. Loehndorf, J.M. Melenk and E.A. Spence, When is the error in the h-bem for solving the Helmholtz equation bounded independently of k? To appear in BIT, M. J. Gander, I. G. Graham and E. A. Spence, How should one choose the shift for the shifted Laplacian to be a good preconditioner for the Helmholtz equation?, submitted I.G. Graham, E.A. Spence and E. Vainikko, Convergence of additive Schwarz methods for the Helmholtz equation with and without absorption, in preparation. S.N. Chandler-Wilde, I.G. Graham, S. Langdon and E.A. Spence, Boundary Integral Equation Methods for High Frequency Scattering Problems Acta Numerica 2012.

11 First problem When is the error in the h version boundary element method bounded independently of k?

12 Fundamental solution for the Helmholtz equation u + k 2 u = 0 G k (x, y) = i 4 H(1) 0 (k x y ) 2D exp(ik x y ) 4π x y 3D Phase: k x y single layer potential : (S k φ)(x) = Γ G k(x, y)φ(y)ds(y), double layer: (D k φ)(x) = Γ [ n(y)g k (x, y)]φ(y)ds(y), adjoint double layer: D k (switch roles of x and y).

13 Combined potential boundary integral formulations Exterior scattering problem with incident field u I : Green s identity for u S in Ω : S k ( n u S + n u I ) D k (u S + u I ) = ( u S + 0) in Ω (1) Limit to boundary Γ: Equation for unknown v := n u but with spurious frequencies. Take normal derivative in (1) and combine with (1): combined potential formulation ( ) 1 R k v := 2 I + D k v iks k v = n u I iku I, or k η Alternative indirect method: ( ) 1 R k φ := 2 I + D k φ iks k φ = u I,

14 Combined potential boundary integral formulations Exterior scattering problem with incident field u I : Green s identity for u I in Ω: S k ( n u S + n u I ) D k (u S + u I ) = ( u S + 0) in Ω (1) Limit to boundary Γ: Equation for unknown v := n u but with spurious frequencies. Take normal derivative in (1) and combine with (1): combined potential formulation ( ) 1 R k v := 2 I + D k v iks k v = n u I iku I, or k η Alternative indirect method: R k φ := ( ) 1 2 I + D k φ iks k φ = u I,

15 Combined potential boundary integral formulations Exterior scattering problem with incident field u I : Green s identity for u I in Ω: S k ( n u S + n u }{{} I ) D k (u S + u I ) = ( u }{{} S + 0) in Ω }{{} (1) nu =0 u I Limit to boundary Γ: Equation for unknown v := n u but with spurious frequencies. Take normal derivative in (1) and combine with rik (1): combined potential formulation ( ) 1 R k v := 2 I + D k v iks k v = n u I iku I, or k η Alternative indirect method: R k φ := ( ) 1 2 I + D k φ iks k φ = u I,

16 Combined potential boundary integral formulations Exterior scattering problem with incident field u I : Green s identity for u I in Ω: S k ( n u S + n u }{{} I ) D k (u S + u I ) = ( u }{{} S + 0) in Ω }{{} (1) nu =0 u I Limit to boundary Γ: Equation for unknown v := n u but with spurious frequencies. Take normal derivative in (1) and combine with ik (1): combined potential formulation ( ) 1 R k v := 2 I + D k v iks k v = n u I iku I, or k η Alternative indirect method: R k φ := ( ) 1 2 I + D k φ iks k φ = u I,

17 Combined potential boundary integral formulations Exterior scattering problem with incident field u I : Green s identity for u I in Ω: S k ( n u S + n u }{{} I ) D k (u S + u I ) = ( u }{{} S + 0) in Ω }{{} (1) nu =0 u I Limit to boundary Γ: Equation for unknown v := n u but with spurious frequencies. Take normal derivative in (1) and combine with ik (1): direct combined potential formulation ( ) 1 R k v := 2 I + D k v iks k v = n u I iku I, or k η Alternative indirect method: R k φ := ( ) 1 2 I + D k φ iks k φ = u I,

18 Combined potential boundary integral formulations Exterior scattering problem with incident field u I : Green s identity for u I in Ω: S k ( n u S + n u }{{} I ) D k (u S + u I ) = ( u }{{} S + 0) in Ω }{{} (1) nu =0 u I Limit to boundary Γ: Equation for unknown v := n u but with spurious frequencies. Take normal derivative in (1) and combine with ik (1): direct combined potential formulation ( ) 1 R k v := 2 I + D k v iks k v = n u I iku I, or k η Alternative indirect method: R k φ := ( ) 1 2 I + D k φ iks k φ = u I,

19 BEM analysis - Classical setting Fredholm integral equations of the Second kind R k v = (λi + L k )v = f k R k φ = (λi + L k )φ = g k (λ = 1/2) Galerkin method in approximating space V N (or V h ). e.g. piecewise polynomials of fixed degree p. Solution v N or φ N, e.g. (λi + P N L k )v N = P N f k

20 BEM analysis - Classical setting Fredholm integral equations of the second kind R k v = (λi + L k )v = f k R k φ = (λi + L k )φ = g k (λ = 1/2) Galerkin method in approximating space V N (or V h ). e.g. piecewise polynomials of fixed degree p. Solution v N or φ N, e.g. (λi + P N L k )v N = P N f k v v N = λ (λi P N L k }{{ ) 1 (v P } N v) }{{} stability best approx

21 Question 1 (best approximation error) When are inf wn V N v w N L 2 (Γ) v L 2 (Γ) and inf wn V N φ w N L 2 (Γ) bounded independently of k? φ L 2 (Γ)

22 Question 2 (quasioptimality) When are v v N L 2 (Γ) inf wn V N v w N L 2 (Γ) and φ φ N L 2 (Γ) bounded independently of k? inf wn V N φ w N L 2 (Γ)

23 If both hold...(bound on relative errors) v v N L 2 (Γ) v L 2 (Γ) and φ φ N L 2 (Γ) φ L 2 (Γ) bounded indpendently of k.

24 Answers: Question 1 ( direct version v = n u) When is inf wn V N v w N L 2 (Γ) bounded independently of k? v L 2 (Γ) Theorem If Ω is C and convex then for h BEM, inf w h V h v w h L 2 (Γ) (hk)p v L 2 (Γ) so hk 1 is sufficient for Question 1.

25 Proof uses v(x) := u/ n(x) = kv (x, k) exp(ikx â), x Γ, Theorem Dominguez, IGG, Smyshlyaev, 2007 { Cn, n = 0, 1, D n V (x, k) C n k 1 (k 1/3 + dist(x, SB)) (n+2) n 2, where SB = {x Γ : n(x).â = 0} shadow boundary. SB â Proves, e.g. v H 1 (Γ) k v L 2 (Γ)

26 Answers: Question 1 ( direct version v = n u) When is inf wn V N v w N L 2 (Γ) bounded independently of k? v L 2 (Γ) Theorem If Ω is a convex polygon then there is a mesh with O(N) points so that, inf w h V h v w h L 2 (Γ) k N v L 2 (Γ) so k/n 1 is sufficient for Question 1. (Requires sup x Ω u(x).)

27 Proof uses: Theorem Chandler-Wilde and Langdon (2007) γ u (s) = 2 ui n n (s) + eiks v + (s) + e iks v (s) where s is distance along γ, and k n v (n) + (s) { Cn (ks) 1/2 n, ks 1, C n (ks) α n, 0 < ks 1, where α < 1/2 depends on the corner angle.

28 Answers: Question 1: Indirect method λφ = L k φ = iks k φ + D k φ To estimate the derivatives of φ: S k H 1 L 2 k (d 1)/2 (Γ Lipschitz) D k H 1 L 2 k (d+1)/2 (Γ smooth enough) These imply φ H 1 (Γ) k (d+1)/2 φ L 2 (Γ) And so hk (d+1)/2 1 is sufficient for Question 1.

29 Answers: Question 2 (classical approach) R k v := (λi + L k )v = f k compact perturbation (λi + P h L k )v h = P h f k Galerkin method Lemma [Atkinson, Anselone, 1960 s...] If (I P h )L k (λi + L k ) 1 << 1, then v v h (λi + L k ) 1 inf w h V h v w h Application: and in addition: (I P h )L k h L k L 2 H1 hk(d+1)/2 (λi + L k ) 1 1 [Chandler-Wilde & Monk, 2008] Lipschitz star-shaped Theorem Hence quasioptimality if hk (d+1)/2 C

30 Tools We used in this talk k explicit bounds on norms of L k, L k (where R k = 1 2 I + L k ), etc. needed smooth enough domains k explicit bounds on inverses (R k ) 1, (R k ) 1 needed Lipschitz star-shaped We will need in the next talk Bound on the solution operator for the Helmhotz BP PDE itself. connection between the two illustrates the role of star-shaped.

31 The Subtlety of Behaviour of L k and R 1 k Equivalently L k and (R k ) 1 L k, R 1 k k 1/3, 1 k 1/2, 1 k 1/2 m, k 7/5 m k 1/2 m, e γkm Chandler-Wilde, IGG et al (2009), Betcke, Chandler-Wilde, IGG et al (2011).

32 Numerical Experiments: domain [0, 0.5] [0, 5] rectangle 1 + γ 2 p v v N L 2 (Γ) inf wn V N v w N L 2 (Γ) norm estimate 10 1 A A 1 O(k 1/2 ) degree p = 0, wavenumber k N k γ 0 γ hk 1

33 Numerical Experiments: trapping domain C shaped domain r r r 3 r r 3 norm estimate 10 1 A A 1 O(k 1/2 ) O(k 0.9 ) wavenumber k m k N γ 0 γ

34 Tools Open question: Prove hk 1 sufficient for quasioptimality We used in this talk k explicit bounds on norms of L k, L k (where R k = 1 2 I + L k ), etc. needed smooth enough domains k explicit bounds on inverses (R k ) 1, (R k ) 1 needed Lipschitz star-shaped We will need in the next talk Bound on the solution operator for the Helmholtz BVP PDE itself. connection between the two illustrates the role of star-shaped.

35 Tools Open question: Prove hk 1 sufficient for quasioptimality We used in this talk k explicit bounds on norms of L k, L k (where R k = 1 2 I + L k ), etc. needed smooth enough domains k explicit bounds on inverses (R k ) 1, (R k ) 1 needed Lipschitz star-shaped We will need in the next talk Bound on the solution operator for the Helmholtz BVP PDE itself. connection between the two illustrates the role of star-shaped.

36 Trapping domains Can things get bad in the non-star-shaped case? Theorem If the exterior domain Ω contains a square Q of side length a and the boundary Γ coincides with two parallel sides of Q, then if 2ak = mπ for any positive integer m, R 1 k (ak)9/10. Ω Q Ω

37 Quasimodes (family of) sources g and solutions v of Helmholtz problem v + k 2 v = g in Ω v = 0 on Γ + Sommerfeld condition, where v L 2 (Ω ) M k g L 2 (Ω ), with M k large. This could contradict the bound v L 2 (Ω ) 1 k f L 2 (Ω ) which holds in star-shaped case (see next lecture). In fact R k ( nv) = ( n v I ikv I ) where v I is the Newtonian potential generated by g implies growth of (R k ) 1 More generally...

38 (R k ) 1 k (d 2) M k O(k (d 1)/2 ) With elliptic cavity M k can increase exponentially. [Betcke, Chandler-Wilde, IGG, Langdon, Lindner, 2011]

39 Final theorem for today... Model interior impedance problem: u k 2 u = f in bounded domain Ω u iku n = g on Γ := Ω...Also truncated sound-soft scattering problems in Ω Theorem (Stability) Assume Ω is Lipschitz and star-shaped. Then, u 2 L 2 (Ω) + k2 u 2 L 2 (Ω) }{{} =: u 2 1,k f 2 L 2 (Ω) + g 2 L 2 (Γ), k [Melenk 95, Cummings & Feng 06] Central result in next lecture

On domain decomposition preconditioners for finite element approximations of the Helmholtz equation using absorption

On domain decomposition preconditioners for finite element approximations of the Helmholtz equation using absorption On domain decomposition preconditioners for finite element approximations of the Helmholtz equation using absorption Ivan Graham and Euan Spence (Bath, UK) Collaborations with: Paul Childs (Emerson Roxar,

More information

Shifted Laplace and related preconditioning for the Helmholtz equation

Shifted Laplace and related preconditioning for the Helmholtz equation Shifted Laplace and related preconditioning for the Helmholtz equation Ivan Graham and Euan Spence (Bath, UK) Collaborations with: Paul Childs (Schlumberger Gould Research), Martin Gander (Geneva) Douglas

More information

When is the error in the h BEM for solving the Helmholtz equation bounded independently of k?

When is the error in the h BEM for solving the Helmholtz equation bounded independently of k? BIT manuscript No. (will be inserted by the editor) When is the error in the h BEM for solving the Helmholtz equation bounded independently of k? I. G. Graham M. Löhndorf J. M. Melenk E. A. Spence Received:

More information

Coercivity of high-frequency scattering problems

Coercivity of high-frequency scattering problems Coercivity of high-frequency scattering problems Valery Smyshlyaev Department of Mathematics, University College London Joint work with: Euan Spence (Bath), Ilia Kamotski (UCL); Comm Pure Appl Math 2015.

More information

High Frequency Scattering by Convex Polygons Stephen Langdon

High Frequency Scattering by Convex Polygons Stephen Langdon Bath, October 28th 2005 1 High Frequency Scattering by Convex Polygons Stephen Langdon University of Reading, UK Joint work with: Simon Chandler-Wilde Steve Arden Funded by: Leverhulme Trust University

More information

Acoustic scattering : high frequency boundary element methods and unified transform methods

Acoustic scattering : high frequency boundary element methods and unified transform methods Acoustic scattering : high frequency boundary element methods and unified transform methods Book or Report Section Accepted Version Chandler Wilde, S.N. and Langdon, S. (2015) Acoustic scattering : high

More information

Keywords: Helmholtz equation, iterative method, preconditioning, high frequency, shifted Laplacian preconditioner, GMRES.

Keywords: Helmholtz equation, iterative method, preconditioning, high frequency, shifted Laplacian preconditioner, GMRES. Noname manuscript No. (will be inserted by the editor) Applying GMRES to the Helmholtz equation with shifted Laplacian preconditioning: What is the largest shift for which wavenumber-independent convergence

More information

Trefftz-Discontinuous Galerkin Methods for Acoustic Scattering - Recent Advances

Trefftz-Discontinuous Galerkin Methods for Acoustic Scattering - Recent Advances Trefftz-Discontinuous Galerkin Methods for Acoustic Scattering - Recent Advances Ilaria Perugia Dipartimento di Matematica Università di Pavia (Italy) In collaboration with Ralf Hiptmair, Christoph Schwab,

More information

High Frequency Scattering by Convex Polygons Stephen Langdon

High Frequency Scattering by Convex Polygons Stephen Langdon High Frequency Scattering by Convex Polygons Stephen Langdon University of Reading, UK Joint work with: Simon Chandler-Wilde, Steve Arden Funded by: Leverhulme Trust University of Maryland, September 2005

More information

High Frequency Solution Behaviour in Wave Scattering Problems Simon Chandler-Wilde University of Reading, UK

High Frequency Solution Behaviour in Wave Scattering Problems Simon Chandler-Wilde University of Reading, UK High Frequency Solution Behaviour in Wave Scattering Problems Simon Chandler-Wilde University of Reading, UK www.reading.ac.uk/~sms03snc With: Steve Langdon, Andrea Moiola (Reading), Ivan Graham, Euan

More information

A GALERKIN BOUNDARY ELEMENT METHOD FOR HIGH FREQUENCY SCATTERING BY CONVEX POLYGONS

A GALERKIN BOUNDARY ELEMENT METHOD FOR HIGH FREQUENCY SCATTERING BY CONVEX POLYGONS A GALERKIN BOUNDARY ELEMENT METHOD FOR HIGH FREQUENCY SCATTERING BY CONVEX POLYGONS S N CHANDLER-WILDE AND S LANGDON Abstract. In this paper we consider the problem of time-harmonic acoustic scattering

More information

Coercivity of Combined Boundary Integral Equations in High-Frequency Scattering

Coercivity of Combined Boundary Integral Equations in High-Frequency Scattering Coercivity of Combined Boundary Integral Equations in High-Frequency Scattering EUAN A. SPENCE University of Bath ILIA V. KAMOTSKI University College London AND VALERY P. SMYSHLYAEV University College

More information

Multiscale methods for time-harmonic acoustic and elastic wave propagation

Multiscale methods for time-harmonic acoustic and elastic wave propagation Multiscale methods for time-harmonic acoustic and elastic wave propagation Dietmar Gallistl (joint work with D. Brown and D. Peterseim) Institut für Angewandte und Numerische Mathematik Karlsruher Institut

More information

Department of Mathematics and Statistics

Department of Mathematics and Statistics School of Mathematical and Physical Sciences Department of Mathematics and Statistics Preprint MPS-11-18 5 November 11 A High Frequency hp Boundary Element Method for Scattering by Convex Polygons by D.P.

More information

arxiv: v1 [math.na] 11 Mar 2019 A high frequency boundary element method for scattering by a class of multiple obstacles

arxiv: v1 [math.na] 11 Mar 2019 A high frequency boundary element method for scattering by a class of multiple obstacles (2019) Page 1 of 39 arxiv:1903.04449v1 [math.na] 11 Mar 2019 A high frequency boundary element method for scattering by a class of multiple obstacles ANDREW GIBBS, DEPT. OF COMPUTER SCIENCE, K.U. LEUVEN,

More information

A High Frequency Boundary Element Method for Scattering by a Class of Nonconvex Obstacles

A High Frequency Boundary Element Method for Scattering by a Class of Nonconvex Obstacles School of Mathematical and Physical Sciences Department of Mathematics and Statistics Preprint MPS-2012-04 28 May 2012 A High Frequency Boundary Element Method for Scattering by a Class of Nonconvex Obstacles

More information

A GALERKIN BOUNDARY ELEMENT METHOD FOR HIGH FREQUENCY SCATTERING BY CONVEX POLYGONS

A GALERKIN BOUNDARY ELEMENT METHOD FOR HIGH FREQUENCY SCATTERING BY CONVEX POLYGONS A GALERKIN BOUNDARY ELEMENT METHOD FOR HIGH FREQUENCY SCATTERING BY CONVEX POLYGONS READING UNIVERSITY NUMERICAL ANALYSIS REPORT 7/06 S N CHANDLER-WILDE AND S LANGDON Abstract. In this paper we consider

More information

Wavenumber-explicit analysis for the Helmholtz h-bem: error estimates and iteration counts for the Dirichlet problem

Wavenumber-explicit analysis for the Helmholtz h-bem: error estimates and iteration counts for the Dirichlet problem Wavenumber-explicit analysis for the Helmholtz h-bem: error estimates and iteration counts for the Dirichlet problem Jeffrey Galkowski, Eike H. Müller, Euan A. Spence June 11, 2018 Abstract We consider

More information

Boundary integral methods in high frequency scattering

Boundary integral methods in high frequency scattering Boundary integral methods in high frequency scattering Simon N. Chandler-Wilde Ivan G. Graham Abstract In this article we review recent progress on the design, analysis and implementation of numerical-asymptotic

More information

The Helmholtz Equation

The Helmholtz Equation The Helmholtz Equation Seminar BEM on Wave Scattering Rene Rühr ETH Zürich October 28, 2010 Outline Steklov-Poincare Operator Helmholtz Equation: From the Wave equation to Radiation condition Uniqueness

More information

Electromagnetic wave propagation. ELEC 041-Modeling and design of electromagnetic systems

Electromagnetic wave propagation. ELEC 041-Modeling and design of electromagnetic systems Electromagnetic wave propagation ELEC 041-Modeling and design of electromagnetic systems EM wave propagation In general, open problems with a computation domain extending (in theory) to infinity not bounded

More information

Acoustic and electromagnetic transmission problems: wavenumber-explicit bounds and resonance-free regions

Acoustic and electromagnetic transmission problems: wavenumber-explicit bounds and resonance-free regions WONAPDE, COMPUTATIONAL ELECTROMAGNETISM MS, CONCEPCIÓN, 21 25 JANUARY 2019 Acoustic and electromagnetic transmission problems: wavenumber-explicit bounds and resonance-free regions Andrea Moiola Joint

More information

arxiv: v1 [math.na] 19 Nov 2018

arxiv: v1 [math.na] 19 Nov 2018 A Bivariate Spline Solution to the Exterior Helmholtz Equation and its Applications Shelvean Kapita Ming-Jun Lai arxiv:1811.07833v1 [math.na] 19 Nov 2018 November 20, 2018 Abstract We explain how to use

More information

A high frequency boundary element method for scattering by a class of nonconvex obstacles

A high frequency boundary element method for scattering by a class of nonconvex obstacles A high frequency boundary element method for scattering by a class of nonconvex obstacles Article Accepted Version Chandler Wilde, S. N., Hewett, D. P., Langdon, S. and Twigger, A. (2015) A high frequency

More information

High Frequency Boundary Element Methods for Scattering by Convex Polygons

High Frequency Boundary Element Methods for Scattering by Convex Polygons High Frequency Boundary Element Methods for Scattering by Convex Polygons J. D. Coleman August 1, 006 1 Acknowledgements I would like to acknowledge the EPSRC for financially supporting this work. I would

More information

When all else fails, integrate by parts an overview of new and old variational formulations for linear elliptic PDEs

When all else fails, integrate by parts an overview of new and old variational formulations for linear elliptic PDEs When all else fails, integrate by parts an overview of new and old variational formulations for linear elliptic PDEs E. A. Spence December 31, 2014 Abstract We give an overview of variational formulations

More information

Two-level Domain decomposition preconditioning for the high-frequency time-harmonic Maxwell equations

Two-level Domain decomposition preconditioning for the high-frequency time-harmonic Maxwell equations Two-level Domain decomposition preconditioning for the high-frequency time-harmonic Maxwell equations Marcella Bonazzoli 2, Victorita Dolean 1,4, Ivan G. Graham 3, Euan A. Spence 3, Pierre-Henri Tournier

More information

Shadow boundary effects in hybrid numerical-asymptotic methods for high frequency scattering

Shadow boundary effects in hybrid numerical-asymptotic methods for high frequency scattering Shadow boundary effects in hybrid numerical-asymptotic methods for high frequency scattering arxiv:1409.3776v [math.na] 3 Jun 015 D. P. Hewett Department of Mathematics and Statistics, University of Reading,

More information

Approximation by plane and circular waves

Approximation by plane and circular waves LMS EPSRC DURHAM SYMPOSIUM, 8 16TH JULY 2014 Building Bridges: Connections and Challenges in Modern Approaches to Numerical PDEs Approximation by plane and circular waves Andrea Moiola DEPARTMENT OF MATHEMATICS

More information

A Galerkin boundary element method for high frequency scattering by convex polygons

A Galerkin boundary element method for high frequency scattering by convex polygons A Galerkin boundary element method for high frequency scattering by convex polygons Article Published Version Chandler Wilde, S. N. and Langdon, S. (2007) A Galerkin boundary element method for high frequency

More information

Solving Time-Harmonic Scattering Problems by the Ultra Weak Variational Formulation

Solving Time-Harmonic Scattering Problems by the Ultra Weak Variational Formulation Introduction Solving Time-Harmonic Scattering Problems by the Ultra Weak Variational Formulation Plane waves as basis functions Peter Monk 1 Tomi Huttunen 2 1 Department of Mathematical Sciences University

More information

Numerical methods for high frequency scattering by multiple obstacles

Numerical methods for high frequency scattering by multiple obstacles Numerical methods for high frequency scattering by multiple obstacles Andrew Gibbs University of Reading A thesis submitted for the degree of Doctor of Philosophy March 2017 Declaration I confirm that

More information

STEKLOFF EIGENVALUES AND INVERSE SCATTERING THEORY

STEKLOFF EIGENVALUES AND INVERSE SCATTERING THEORY STEKLOFF EIGENVALUES AND INVERSE SCATTERING THEORY David Colton, Shixu Meng, Peter Monk University of Delaware Fioralba Cakoni Rutgers University Research supported by AFOSR Grant FA 9550-13-1-0199 Scattering

More information

A two-level domain-decomposition preconditioner for the time-harmonic Maxwell s equations

A two-level domain-decomposition preconditioner for the time-harmonic Maxwell s equations A two-level domain-decomposition preconditioner for the time-harmonic Maxwell s equations Marcella Bonazzoli, Victorita Dolean, Ivan Graham, Euan Spence, Pierre-Henri Tournier To cite this version: Marcella

More information

1 domain-based methods: convergence analysis for hp-fem. 2 discussion of some non-standard FEMs. 3 BEM for Helmholtz problems.

1 domain-based methods: convergence analysis for hp-fem. 2 discussion of some non-standard FEMs. 3 BEM for Helmholtz problems. outline Helmholtz problems at large wavenumber k joint work with S. Sauter (Zürich) M. Löhndorf (Vienna) S. Langdon (Reading) 1 domain-based methods: convergence analysis for hp-fem 2 discussion of some

More information

Lecture 8: Boundary Integral Equations

Lecture 8: Boundary Integral Equations CBMS Conference on Fast Direct Solvers Dartmouth College June 23 June 27, 2014 Lecture 8: Boundary Integral Equations Gunnar Martinsson The University of Colorado at Boulder Research support by: Consider

More information

Is the Helmholtz Equation Really Sign-Indefinite?

Is the Helmholtz Equation Really Sign-Indefinite? SIAM REVIEW Vol. 56, No. 2, pp. 274 312 c 2014 Society for Industrial and Applied Mathematics Is the Helmholtz Equation Really Sign-Indefinite? Andrea Moiola Euan A. Spence Abstract. The usual variational

More information

arxiv: v1 [math.na] 11 Jul 2018

arxiv: v1 [math.na] 11 Jul 2018 An Additive Overlapping Domain Decomposition Method for the Helmholtz Equation Wei Leng Lili Ju arxiv:1807.04180v1 [math.na] 11 Jul 2018 July 12, 2018 Abstract In this paper, we propose and analyze an

More information

The Factorization Method for Inverse Scattering Problems Part I

The Factorization Method for Inverse Scattering Problems Part I The Factorization Method for Inverse Scattering Problems Part I Andreas Kirsch Madrid 2011 Department of Mathematics KIT University of the State of Baden-Württemberg and National Large-scale Research Center

More information

Scientific Computing WS 2017/2018. Lecture 18. Jürgen Fuhrmann Lecture 18 Slide 1

Scientific Computing WS 2017/2018. Lecture 18. Jürgen Fuhrmann Lecture 18 Slide 1 Scientific Computing WS 2017/2018 Lecture 18 Jürgen Fuhrmann juergen.fuhrmann@wias-berlin.de Lecture 18 Slide 1 Lecture 18 Slide 2 Weak formulation of homogeneous Dirichlet problem Search u H0 1 (Ω) (here,

More information

Domain Decomposition Algorithms for an Indefinite Hypersingular Integral Equation in Three Dimensions

Domain Decomposition Algorithms for an Indefinite Hypersingular Integral Equation in Three Dimensions Domain Decomposition Algorithms for an Indefinite Hypersingular Integral Equation in Three Dimensions Ernst P. Stephan 1, Matthias Maischak 2, and Thanh Tran 3 1 Institut für Angewandte Mathematik, Leibniz

More information

Factorization method in inverse

Factorization method in inverse Title: Name: Affil./Addr.: Factorization method in inverse scattering Armin Lechleiter University of Bremen Zentrum für Technomathematik Bibliothekstr. 1 28359 Bremen Germany Phone: +49 (421) 218-63891

More information

Trefftz-discontinuous Galerkin methods for time-harmonic wave problems

Trefftz-discontinuous Galerkin methods for time-harmonic wave problems Trefftz-discontinuous Galerkin methods for time-harmonic wave problems Ilaria Perugia Dipartimento di Matematica - Università di Pavia (Italy) http://www-dimat.unipv.it/perugia Joint work with Ralf Hiptmair,

More information

Technische Universität Graz

Technische Universität Graz Technische Universität Graz A note on the stable coupling of finite and boundary elements O. Steinbach Berichte aus dem Institut für Numerische Mathematik Bericht 2009/4 Technische Universität Graz A

More information

A WAVENUMBER INDEPENDENT BOUNDARY ELEMENT METHOD FOR AN ACOUSTIC SCATTERING PROBLEM

A WAVENUMBER INDEPENDENT BOUNDARY ELEMENT METHOD FOR AN ACOUSTIC SCATTERING PROBLEM A WAVENUMBER INDEPENDENT BOUNDARY ELEMENT METHOD FOR AN ACOUSTIC SCATTERING PROBLEM S LANGDON AND S N CHANDLER-WILDE Abstract. In this paper we consider the impedance boundary value problem for the Helmholtz

More information

Preconditioned space-time boundary element methods for the heat equation

Preconditioned space-time boundary element methods for the heat equation W I S S E N T E C H N I K L E I D E N S C H A F T Preconditioned space-time boundary element methods for the heat equation S. Dohr and O. Steinbach Institut für Numerische Mathematik Space-Time Methods

More information

Trefftz-Discontinuous Galerkin Methods for the Time-Harmonic Maxwell Equations

Trefftz-Discontinuous Galerkin Methods for the Time-Harmonic Maxwell Equations Trefftz-Discontinuous Galerkin Methods for the Time-Harmonic Maxwell Equations Ilaria Perugia Dipartimento di Matematica - Università di Pavia (Italy) http://www-dimat.unipv.it/perugia Joint work with

More information

A coupled BEM and FEM for the interior transmission problem

A coupled BEM and FEM for the interior transmission problem A coupled BEM and FEM for the interior transmission problem George C. Hsiao, Liwei Xu, Fengshan Liu, Jiguang Sun Abstract The interior transmission problem (ITP) is a boundary value problem arising in

More information

Trefftz-Discontinuous Galerkin Methods for Maxwell s Equations

Trefftz-Discontinuous Galerkin Methods for Maxwell s Equations Trefftz-Discontinuous Galerkin Methods for Maxwell s Equations Ilaria Perugia Faculty of Mathematics, University of Vienna Vienna P D E Ralf Hiptmair (ETH Zürich), Andrea Moiola (University of Reading)

More information

Inverse Obstacle Scattering

Inverse Obstacle Scattering , Göttingen AIP 2011, Pre-Conference Workshop Texas A&M University, May 2011 Scattering theory Scattering theory is concerned with the effects that obstacles and inhomogenities have on the propagation

More information

Standard Finite Elements and Weighted Regularization

Standard Finite Elements and Weighted Regularization Standard Finite Elements and Weighted Regularization A Rehabilitation Martin COSTABEL & Monique DAUGE Institut de Recherche MAthématique de Rennes http://www.maths.univ-rennes1.fr/~dauge Slides of the

More information

Boundary integral equations on unbounded rough surfaces: Fredholmness and the finite section method

Boundary integral equations on unbounded rough surfaces: Fredholmness and the finite section method Boundary integral equations on unbounded rough surfaces: Fredholmness and the finite section method Simon N. Chandler-Wilde and Marko Lindner ABSTRACT. We consider a class of boundary integral equations

More information

A high frequency $hp$ boundary element method for scattering by convex polygons

A high frequency $hp$ boundary element method for scattering by convex polygons A high frequency $hp$ boundary element method for scattering by convex polygons Article Published Version Hewett, D. P., Langdon, S. and Melenk, J. M. (213) A high frequency $hp$ boundary element method

More information

arxiv: v2 [physics.comp-ph] 26 Aug 2014

arxiv: v2 [physics.comp-ph] 26 Aug 2014 On-surface radiation condition for multiple scattering of waves Sebastian Acosta Computational and Applied Mathematics, Rice University, Houston, TX Department of Pediatric Cardiology, Baylor College of

More information

Finite difference method for elliptic problems: I

Finite difference method for elliptic problems: I Finite difference method for elliptic problems: I Praveen. C praveen@math.tifrbng.res.in Tata Institute of Fundamental Research Center for Applicable Mathematics Bangalore 560065 http://math.tifrbng.res.in/~praveen

More information

Isogeometric Analysis:

Isogeometric Analysis: Isogeometric Analysis: some approximation estimates for NURBS L. Beirao da Veiga, A. Buffa, Judith Rivas, G. Sangalli Euskadi-Kyushu 2011 Workshop on Applied Mathematics BCAM, March t0th, 2011 Outline

More information

Numerical Solutions to Partial Differential Equations

Numerical Solutions to Partial Differential Equations Numerical Solutions to Partial Differential Equations Zhiping Li LMAM and School of Mathematical Sciences Peking University The Residual and Error of Finite Element Solutions Mixed BVP of Poisson Equation

More information

On complex shifted Laplace preconditioners for the vector Helmholtz equation

On complex shifted Laplace preconditioners for the vector Helmholtz equation On complex shifted Laplace preconditioners for the vector Helmholtz equation C. Vuik, Y.A. Erlangga, M.B. van Gijzen, C.W. Oosterlee, D. van der Heul Delft Institute of Applied Mathematics c.vuik@tudelft.nl

More information

Scientific Computing WS 2018/2019. Lecture 15. Jürgen Fuhrmann Lecture 15 Slide 1

Scientific Computing WS 2018/2019. Lecture 15. Jürgen Fuhrmann Lecture 15 Slide 1 Scientific Computing WS 2018/2019 Lecture 15 Jürgen Fuhrmann juergen.fuhrmann@wias-berlin.de Lecture 15 Slide 1 Lecture 15 Slide 2 Problems with strong formulation Writing the PDE with divergence and gradient

More information

Fast Multipole BEM for Structural Acoustics Simulation

Fast Multipole BEM for Structural Acoustics Simulation Fast Boundary Element Methods in Industrial Applications Fast Multipole BEM for Structural Acoustics Simulation Matthias Fischer and Lothar Gaul Institut A für Mechanik, Universität Stuttgart, Germany

More information

Inverse wave scattering problems: fast algorithms, resonance and applications

Inverse wave scattering problems: fast algorithms, resonance and applications Inverse wave scattering problems: fast algorithms, resonance and applications Wagner B. Muniz Department of Mathematics Federal University of Santa Catarina (UFSC) w.b.muniz@ufsc.br III Colóquio de Matemática

More information

Finite Element Analysis of Acoustic Scattering

Finite Element Analysis of Acoustic Scattering Frank Ihlenburg Finite Element Analysis of Acoustic Scattering With 88 Illustrations Springer Contents Preface vii 1 The Governing Equations of Time-Harmonic Wave Propagation, 1 1.1 Acoustic Waves 1 1.1.1

More information

A FAST SOLVER FOR ELLIPTIC EQUATIONS WITH HARMONIC COEFFICIENT APPROXIMATIONS

A FAST SOLVER FOR ELLIPTIC EQUATIONS WITH HARMONIC COEFFICIENT APPROXIMATIONS Proceedings of ALGORITMY 2005 pp. 222 229 A FAST SOLVER FOR ELLIPTIC EQUATIONS WITH HARMONIC COEFFICIENT APPROXIMATIONS ELENA BRAVERMAN, MOSHE ISRAELI, AND ALEXANDER SHERMAN Abstract. Based on a fast subtractional

More information

Additive Schwarz method for scattering problems using the PML method at interfaces

Additive Schwarz method for scattering problems using the PML method at interfaces Additive Schwarz method for scattering problems using the PML method at interfaces Achim Schädle 1 and Lin Zschiedrich 2 1 Zuse Institute, Takustr. 7, 14195 Berlin, Germany schaedle@zib.de 2 Zuse Institute,

More information

u(0) = u 0, u(1) = u 1. To prove what we want we introduce a new function, where c = sup x [0,1] a(x) and ɛ 0:

u(0) = u 0, u(1) = u 1. To prove what we want we introduce a new function, where c = sup x [0,1] a(x) and ɛ 0: 6. Maximum Principles Goal: gives properties of a solution of a PDE without solving it. For the two-point boundary problem we shall show that the extreme values of the solution are attained on the boundary.

More information

A far-field based T-matrix method for three dimensional acoustic scattering

A far-field based T-matrix method for three dimensional acoustic scattering ANZIAM J. 50 (CTAC2008) pp.c121 C136, 2008 C121 A far-field based T-matrix method for three dimensional acoustic scattering M. Ganesh 1 S. C. Hawkins 2 (Received 14 August 2008; revised 4 October 2008)

More information

A frequency-independent boundary element method for scattering by two-dimensional screens and apertures

A frequency-independent boundary element method for scattering by two-dimensional screens and apertures A frequency-independent boundary element method for scattering by two-dimensional screens and apertures arxiv:1401.2786v2 [math.na] 11 Aug 2014 D. P. Hewett, S. Langdon and S. N. Chandler-Wilde Department

More information

Variational Formulations

Variational Formulations Chapter 2 Variational Formulations In this chapter we will derive a variational (or weak) formulation of the elliptic boundary value problem (1.4). We will discuss all fundamental theoretical results that

More information

Wavenumber explicit convergence analysis for Galerkin discretizations of the Helmholtz equation

Wavenumber explicit convergence analysis for Galerkin discretizations of the Helmholtz equation Zurich Open Repository and Archive University of Zurich Main Library Strickhofstrasse 39 CH-8057 Zurich www.zora.uzh.ch Year: 2011 Wavenumber explicit convergence analysis for Galerkin discretizations

More information

The Mortar Boundary Element Method

The Mortar Boundary Element Method The Mortar Boundary Element Method A Thesis submitted for the degree of Doctor of Philosophy by Martin Healey School of Information Systems, Computing and Mathematics Brunel University March 2010 Abstract

More information

Sobolev spaces, Trace theorems and Green s functions.

Sobolev spaces, Trace theorems and Green s functions. Sobolev spaces, Trace theorems and Green s functions. Boundary Element Methods for Waves Scattering Numerical Analysis Seminar. Orane Jecker October 21, 2010 Plan Introduction 1 Useful definitions 2 Distributions

More information

ON THE INFLUENCE OF THE WAVENUMBER ON COMPRESSION IN A WAVELET BOUNDARY ELEMENT METHOD FOR THE HELMHOLTZ EQUATION

ON THE INFLUENCE OF THE WAVENUMBER ON COMPRESSION IN A WAVELET BOUNDARY ELEMENT METHOD FOR THE HELMHOLTZ EQUATION INTERNATIONAL JOURNAL OF NUMERICAL ANALYSIS AND MODELING Volume 4, Number 1, Pages 48 62 c 2007 Institute for Scientific Computing and Information ON THE INFLUENCE OF THE WAVENUMBER ON COMPRESSION IN A

More information

Estimation of transmission eigenvalues and the index of refraction from Cauchy data

Estimation of transmission eigenvalues and the index of refraction from Cauchy data Estimation of transmission eigenvalues and the index of refraction from Cauchy data Jiguang Sun Abstract Recently the transmission eigenvalue problem has come to play an important role and received a lot

More information

Introduction to finite element exterior calculus

Introduction to finite element exterior calculus Introduction to finite element exterior calculus Ragnar Winther CMA, University of Oslo Norway Why finite element exterior calculus? Recall the de Rham complex on the form: R H 1 (Ω) grad H(curl, Ω) curl

More information

Fast and accurate methods for the discretization of singular integral operators given on surfaces

Fast and accurate methods for the discretization of singular integral operators given on surfaces Fast and accurate methods for the discretization of singular integral operators given on surfaces James Bremer University of California, Davis March 15, 2018 This is joint work with Zydrunas Gimbutas (NIST

More information

Fast Fourier Transform Solvers and Preconditioners for Quadratic Spline Collocation

Fast Fourier Transform Solvers and Preconditioners for Quadratic Spline Collocation Fast Fourier Transform Solvers and Preconditioners for Quadratic Spline Collocation Christina C. Christara and Kit Sun Ng Department of Computer Science University of Toronto Toronto, Ontario M5S 3G4,

More information

Wavenumber-explicit hp-bem for High Frequency Scattering

Wavenumber-explicit hp-bem for High Frequency Scattering ASC Report No. 2/2010 Wavenumber-explicit hp-bem for High Frequency Scattering Maike Löhndorf, Jens Markus Melenk Institute for Analysis and Scientific Computing Vienna University of Technology TU Wien

More information

Chapter 6. Finite Element Method. Literature: (tiny selection from an enormous number of publications)

Chapter 6. Finite Element Method. Literature: (tiny selection from an enormous number of publications) Chapter 6 Finite Element Method Literature: (tiny selection from an enormous number of publications) K.J. Bathe, Finite Element procedures, 2nd edition, Pearson 2014 (1043 pages, comprehensive). Available

More information

Adaptive Boundary Element Methods Part 2: ABEM. Dirk Praetorius

Adaptive Boundary Element Methods Part 2: ABEM. Dirk Praetorius CENTRAL School on Analysis and Numerics for PDEs November 09-12, 2015 Adaptive Boundary Element Methods Part 2: ABEM Dirk Praetorius TU Wien Institute for Analysis and Scientific Computing Outline 1 Boundary

More information

Wave number explicit bounds in timeharmonic

Wave number explicit bounds in timeharmonic Wave number explicit bounds in timeharmonic scattering Article Accepted Version Chandler Wilde, S.N. and Monk, P. 008) Wave numberexplicit bounds in time harmonic scattering. SIAM Journal on Mathematical

More information

Maximum norm estimates for energy-corrected finite element method

Maximum norm estimates for energy-corrected finite element method Maximum norm estimates for energy-corrected finite element method Piotr Swierczynski 1 and Barbara Wohlmuth 1 Technical University of Munich, Institute for Numerical Mathematics, piotr.swierczynski@ma.tum.de,

More information

Numerical solution to the Complex 2D Helmholtz Equation based on Finite Volume Method with Impedance Boundary Conditions

Numerical solution to the Complex 2D Helmholtz Equation based on Finite Volume Method with Impedance Boundary Conditions Open Phys. 016; 14:436 443 Research Article Open Access Angela Handlovičová and Izabela Riečanová* Numerical solution to the Complex D Helmholtz Equation based on Finite Volume Method with Impedance Boundary

More information

A robust computational method for the Schrödinger equation cross sections using an MG-Krylov scheme

A robust computational method for the Schrödinger equation cross sections using an MG-Krylov scheme A robust computational method for the Schrödinger equation cross sections using an MG-Krylov scheme International Conference On Preconditioning Techniques For Scientific And Industrial Applications 17-19

More information

Numerical Solution I

Numerical Solution I Numerical Solution I Stationary Flow R. Kornhuber (FU Berlin) Summerschool Modelling of mass and energy transport in porous media with practical applications October 8-12, 2018 Schedule Classical Solutions

More information

Integral Representation Formula, Boundary Integral Operators and Calderón projection

Integral Representation Formula, Boundary Integral Operators and Calderón projection Integral Representation Formula, Boundary Integral Operators and Calderón projection Seminar BEM on Wave Scattering Franziska Weber ETH Zürich October 22, 2010 Outline Integral Representation Formula Newton

More information

Inverse scattering problem from an impedance obstacle

Inverse scattering problem from an impedance obstacle Inverse Inverse scattering problem from an impedance obstacle Department of Mathematics, NCKU 5 th Workshop on Boundary Element Methods, Integral Equations and Related Topics in Taiwan NSYSU, October 4,

More information

From the Boundary Element Domain Decomposition Methods to Local Trefftz Finite Element Methods on Polyhedral Meshes

From the Boundary Element Domain Decomposition Methods to Local Trefftz Finite Element Methods on Polyhedral Meshes From the Boundary Element Domain Decomposition Methods to Local Trefftz Finite Element Methods on Polyhedral Meshes Dylan Copeland 1, Ulrich Langer 2, and David Pusch 3 1 Institute of Computational Mathematics,

More information

Migration with Implicit Solvers for the Time-harmonic Helmholtz

Migration with Implicit Solvers for the Time-harmonic Helmholtz Migration with Implicit Solvers for the Time-harmonic Helmholtz Yogi A. Erlangga, Felix J. Herrmann Seismic Laboratory for Imaging and Modeling, The University of British Columbia {yerlangga,fherrmann}@eos.ubc.ca

More information

A posteriori error estimates in FEEC for the de Rham complex

A posteriori error estimates in FEEC for the de Rham complex A posteriori error estimates in FEEC for the de Rham complex Alan Demlow Texas A&M University joint work with Anil Hirani University of Illinois Urbana-Champaign Partially supported by NSF DMS-1016094

More information

Computational method for acoustic wave focusing

Computational method for acoustic wave focusing Int. J. Computing Science and Mathematics Vol. 1 No. 1 7 1 Computational method for acoustic wave focusing A.G. Ramm* Mathematics epartment Kansas State University Manhattan KS 6656-6 USA E-mail: ramm@math.ksu.edu

More information

Your first day at work MATH 806 (Fall 2015)

Your first day at work MATH 806 (Fall 2015) Your first day at work MATH 806 (Fall 2015) 1. Let X be a set (with no particular algebraic structure). A function d : X X R is called a metric on X (and then X is called a metric space) when d satisfies

More information

From the Boundary Element DDM to local Trefftz Finite Element Methods on Polyhedral Meshes

From the Boundary Element DDM to local Trefftz Finite Element Methods on Polyhedral Meshes www.oeaw.ac.at From the Boundary Element DDM to local Trefftz Finite Element Methods on Polyhedral Meshes D. Copeland, U. Langer, D. Pusch RICAM-Report 2008-10 www.ricam.oeaw.ac.at From the Boundary Element

More information

A Domain Decomposition Method for Quasilinear Elliptic PDEs Using Mortar Finite Elements

A Domain Decomposition Method for Quasilinear Elliptic PDEs Using Mortar Finite Elements W I S S E N T E C H N I K L E I D E N S C H A F T A Domain Decomposition Method for Quasilinear Elliptic PDEs Using Mortar Finite Elements Matthias Gsell and Olaf Steinbach Institute of Computational Mathematics

More information

Multigrid Methods for Maxwell s Equations

Multigrid Methods for Maxwell s Equations Multigrid Methods for Maxwell s Equations Jintao Cui Institute for Mathematics and Its Applications University of Minnesota Outline Nonconforming Finite Element Methods for a Two Dimensional Curl-Curl

More information

Technische Universität Graz

Technische Universität Graz Technische Universität Graz Stability of the Laplace single layer boundary integral operator in Sobolev spaces O. Steinbach Berichte aus dem Institut für Numerische Mathematik Bericht 2016/2 Technische

More information

Scattering, Clouds & Climate: A Short Workshop of Exploration Mathematical Institute, University of Oxford 24 th - 25 th March 2014

Scattering, Clouds & Climate: A Short Workshop of Exploration Mathematical Institute, University of Oxford 24 th - 25 th March 2014 Scattering, Clouds & Climate: A Short Workshop of Exploration Mathematical Institute, University of Oxford 24 th - 25 th March 2014 A two-day workshop bringing together mathematicians and atmospheric physicists

More information

The elliptic sinh-gordon equation in the half plane

The elliptic sinh-gordon equation in the half plane Available online at www.tjnsa.com J. Nonlinear Sci. Appl. 8 25), 63 73 Research Article The elliptic sinh-gordon equation in the half plane Guenbo Hwang Department of Mathematics, Daegu University, Gyeongsan

More information

Mesh Grading towards Singular Points Seminar : Elliptic Problems on Non-smooth Domain

Mesh Grading towards Singular Points Seminar : Elliptic Problems on Non-smooth Domain Mesh Grading towards Singular Points Seminar : Elliptic Problems on Non-smooth Domain Stephen Edward Moore Johann Radon Institute for Computational and Applied Mathematics Austrian Academy of Sciences,

More information

LECTURE 5 APPLICATIONS OF BDIE METHOD: ACOUSTIC SCATTERING BY INHOMOGENEOUS ANISOTROPIC OBSTACLES DAVID NATROSHVILI

LECTURE 5 APPLICATIONS OF BDIE METHOD: ACOUSTIC SCATTERING BY INHOMOGENEOUS ANISOTROPIC OBSTACLES DAVID NATROSHVILI LECTURE 5 APPLICATIONS OF BDIE METHOD: ACOUSTIC SCATTERING BY INHOMOGENEOUS ANISOTROPIC OBSTACLES DAVID NATROSHVILI Georgian Technical University Tbilisi, GEORGIA 0-0 1. Formulation of the corresponding

More information

AposteriorierrorestimatesinFEEC for the de Rham complex

AposteriorierrorestimatesinFEEC for the de Rham complex AposteriorierrorestimatesinFEEC for the de Rham complex Alan Demlow Texas A&M University joint work with Anil Hirani University of Illinois Urbana-Champaign Partially supported by NSF DMS-1016094 and a

More information