First kind boundary integral formulation for the Hodge-Helmholtz equation
|
|
- Emmeline Poole
- 5 years ago
- Views:
Transcription
1 First kind boundary integral formulation for the Hodge-Helmholtz equation X.Claeys & R.Hiptmair LJLL UPMC / INRIA Alpines, SAM ETH Zürich
2 Integral equation for low frequency Maxwell Ω = bounded Lipschitz, = Ω Find E H(curl,Ω) such that curl(curl E) κ 2 E = 0 in Ω n (E n) = g on. n j Ω j j Boundary integral formulation? at low frequency? Electric field integral equation (EFIE) : find u À 1/2 ( Ú,) such that G κ(x y) ( κ 2 Ú v(x) Ú u(y) v(x) u(y) ) dσ(x, y) = g(x) v(x)dσ(x) v À 1/2 ( Ú,)
3 Integral equation for low frequency Maxwell Ω = bounded Lipschitz, = Ω Find E H(curl,Ω) such that curl(curl E) κ 2 E = 0 in Ω n (E n) = g on. n j Ω j j Boundary integral formulation? at low frequency? Electric field integral equation (EFIE) : find u À 1/2 ( Ú,) such that G κ(x y) ( κ 2 Ú v(x) Ú u(y) v(x) u(y) ) dσ(x, y) = g(x) v(x)dσ(x) v À 1/2 ( Ú,) Green kernel : G κ(x) := exp(ıκ x )/(4π x )
4 Integral equation for low frequency Maxwell Ω = bounded Lipschitz, = Ω Find E H(curl,Ω) such that curl(curl E) κ 2 E = 0 in Ω n (E n) = g on. n j Ω j j Boundary integral formulation? at low frequency? Electric field integral equation (EFIE) : find u À 1/2 ( Ú,) such that G κ(x y) ( κ 2 Ú v(x) Ú u(y) v(x) u(y) ) dσ(x, y) = g(x) v(x)dσ(x) v À 1/2 ( Ú,) Low frequency breakdown
5 Low frequency breakdown Loop-star/tree stabilisation : Wilton & Glisson (1981), Vecchi (1999), Zhao & Chew (2000), Lee & Burkholder (2003), Eibert (2004), Andriulli (2012). Debye sources : Greengard & Epstein (2010), Greengard, Epstein & O Neil (2013 & 2015), Vico, Ferrando, Greengard & Gimbutas (2016). Current and charge formulation : Taskinen & Ylä-Oijala (2006) Taskinen & Vanska (2007), Taskinen (2009), Bendali & al. (2012), Vico & al. (2013), Ganesh, Hawkins & Volkov (2014).
6 Low frequency breakdown Loop-star/tree stabilisation : Wilton & Glisson (1981), Vecchi (1999), Zhao & Chew (2000), Lee & Burkholder (2003), Eibert (2004), Andriulli (2012). Debye sources : Greengard & Epstein (2010), Greengard, Epstein & O Neil (2013 & 2015), Vico, Ferrando, Greengard & Gimbutas (2016). Current and charge formulation : Taskinen & Ylä-Oijala (2006) Taskinen & Vanska (2007), Taskinen (2009), Bendali & al. (2012), Vico & al. (2013), Ganesh, Hawkins & Volkov (2014). Current and charge formulations have been shown to stem from Maxwell s equations by relaxing the contraint associated to divergence by means of Lagrange multipliers. The later formulation of electromagnetics is known as Picard s system (introduced by Picard, 1984). Goal of this work : clarifying the analysis of this approach (Calderón calculus? traces? well-posedness? etc...) Initial idea : rewrite Picard s system as a 2nd order problem and try to adapt the analysis presented in Costabel (1988).
7 Outline I.II Maxwell as vector Helmholtz equation II.I Hodge-Helmholtz potential theory III. Low frequency regime
8 Outline I.II Maxwell as vector Helmholtz equation II.I Hodge-Helmholtz potential theory III. Low frequency regime
9 Reformulation of Maxwell s BVP Proposition (Hazard & Lenoir, 1996) : Let Ω R 3 be bounded Lipschitz, and assume that κ 2 / S( Ö), then for any g H 1/2 (curl,), we have u H(curl,Ω) such that curl 2 u κ 2 u = 0 in Ω n (u n) = g on u X(Ω) := H(curl,Ω) H( Ú,Ω) curl 2 u ( Ú u) κ 2 u = 0 in Ω n (u n) = g on ( Ú u) = 0 on
10 Reformulation of Maxwell s BVP Proposition (Hazard & Lenoir, 1996) : Let Ω R 3 be bounded Lipschitz, and assume that κ 2 / S( Ö), then for any g H 1/2 (curl,), we have u H(curl,Ω) such that curl 2 u κ 2 u = 0 in Ω n (u n) = g on u X(Ω) := H(curl,Ω) H( Ú,Ω) curl 2 u ( Ú u) κ 2 u = 0 in Ω n (u n) = g on ( Ú u) = 0 on = u Hodge-Laplace operator
11 Reformulation of Maxwell s BVP Proposition (Hazard & Lenoir, 1996) : Let Ω R 3 be bounded Lipschitz, and assume that κ 2 / S( Ö), then for any g H 1/2 (curl,), we have u H(curl,Ω) such that curl 2 u κ 2 u = 0 in Ω n (u n) = g on Ω u X(Ω) := H(curl,Ω) H( Ú,Ω) curl 2 u ( Ú u) κ 2 u = 0 in Ω n (u n) = g on ( Ú u) = 0 on = u Green s formula Hodge-Laplace operator curl 2 (u) v u curl 2 (v) dx = v (n curl u) u (n curl v) dσ Ω ( Úv) u v ( Úu) dx = Ú(v) n u Ú(u) n v dσ
12 Reformulation of Maxwell s BVP + = Proposition (Hazard & Lenoir, 1996) : Let Ω R 3 be bounded Lipschitz, and assume that κ 2 / S( Ö), then for any g H 1/2 (curl,), we have u H(curl,Ω) such that curl 2 u κ 2 u = 0 in Ω n (u n) = g on Ω u X(Ω) := H(curl,Ω) H( Ú,Ω) curl 2 u ( Ú u) κ 2 u = 0 in Ω n (u n) = g on ( Ú u) = 0 on = u Green s formula Hodge-Laplace operator curl 2 (u) v u curl 2 (v) dx = v (n curl u) u (n curl v) dσ Ω Ω ( Úv) u v ( Úu) dx = Ú(v) n u Ú(u) n v dσ u v v u dx
13 Reformulation of Maxwell s BVP Proposition (Hazard & Lenoir, 1996) : Let Ω R 3 be bounded Lipschitz, and assume that κ 2 / S( Ö), then for any g H 1/2 (curl,), we have u H(curl,Ω) such that curl 2 u κ 2 u = 0 in Ω n (u n) = g on Ω u X(Ω) := H(curl,Ω) H( Ú,Ω) curl 2 u ( Ú u) κ 2 u = 0 in Ω n (u n) = g on ( Ú u) = 0 on = u Green s formula Hodge-Laplace operator u v v u dx = v (n curl u) u (n curl v) dσ Ω ( Úv) u v ( Úu) dx + Ú(v) n u Ú(u) n v dσ
14 Reformulation of Maxwell s BVP Proposition (Hazard & Lenoir, 1996) : Let Ω R 3 be bounded Lipschitz, and assume that κ 2 / S( Ö), then for any g H 1/2 (curl,), we have u H(curl,Ω) such that curl 2 u κ 2 u = 0 in Ω n (u n) = g on Ω u X(Ω) := H(curl,Ω) H( Ú,Ω) curl 2 u ( Ú u) κ 2 u = 0 in Ω n (u n) = g on ( Ú u) = 0 on = u Green s formula Hodge-Laplace operator u v v u dx = v (n curl u) u (n curl v) dσ Ω ( Úv) u v ( Úu) dx + Ú(v) n u Ú(u) n v dσ. "Neumann trace" T n(u) := (n curl(u), n u). "Dirichlet trace" T d(u) := (n u n, Ú(u))
15 Green s formula for the Hodge Laplacian u v v u dx = T d(v) T n(u) T d(u) T n(v) dσ Ω Û Ö T d(u) := (n u n, Ú(u) ) Û Ö T n(u) := (n curl(u ), n u ) Proposition Denote X(,Ω) := {u X(Ω), curl 2 (u) Ä 2 (Ω) 3, ( Ú u) Ä 2 (Ω) 3 }. Then the following trace operators are continuous, surjective, and admit a continuous right inverse, T d : X(,Ω) H d() := À 1 2 ( Ú,) À () T n : X(,Ω) H n() := À 1 2(curl,) À 1 2() This choice for for Dirichlet and Neumann traces is apparently the only one that garantees good continuity/surjectivity properties. It was also considered in [Mitrea& al, 2016] and [Schwarz, 1995].
16 Green s formula for the Hodge Laplacian u v v u dx = T d(v) T n(u) T d(u) T n(v) dσ Ω Û Ö T d(u) := (n u n, Ú(u) ) Û Ö T n(u) := (n curl(u ), n u ) Proposition Denote X(,Ω) := {u X(Ω), curl 2 (u) Ä 2 (Ω) 3, ( Ú u) Ä 2 (Ω) 3 }. Then the following trace operators are continuous, surjective, and admit a continuous right inverse, T d : X(,Ω) H d() := À 1 2 ( Ú,) À () T n : X(,Ω) H n() := À 1 2(curl,) À 1 2() Important : Td(v) Tn(u) Ω curl(u) curl(v)+ Ú(u) Ú(v)dx ± Ω v udx which makes using Costabel s variational analysis (apparently) impossible.
17 Outline I.II Maxwell as vector Helmholtz equation II.I Hodge-Helmholtz potential theory III. Low frequency regime
18 Repesentation formula Green s formula may be rewritten as 1 Ω 1 Ω = T dt n T nt d in the sense of distributions. Hence for any u X(Ω) satisfying u +κ 2 u = 0 in Ω, we have G κ ( ( +κ 2 )1 Ω u = T nt d(u) T dt n(u) ) Û Ö G κ(x) := exp(ıκ x )/(4π x )
19 Repesentation formula Green s formula may be rewritten as 1 Ω 1 Ω = T dt n T nt d in the sense of distributions. Hence for any u X(Ω) satisfying u +κ 2 u = 0 in Ω, we have G κ ( ( +κ 2 )1 Ω u = T nt d(u) T dt n(u) ) Û Ö G κ(x) := exp(ıκ x )/(4π x )
20 Repesentation formula Green s formula may be rewritten as 1 Ω 1 Ω = T dt n T nt d in the sense of distributions. Hence for any u X(Ω) satisfying u +κ 2 u = 0 in Ω, we have DL κ(t d(u))(x)+sl κ(t n(u))(x) = 1 Ω (x)u(x), x R 3 Û Ö Û Ö DL κ := +G κ T n : H d() X(,Ω) SL κ := G κ T d : H n() X(,Ω)
21 Repesentation formula Green s formula may be rewritten as 1 Ω 1 Ω = T dt n T nt d in the sense of distributions. Hence for any u X(Ω) satisfying u +κ 2 u = 0 in Ω, we have DL κ(t d(u))(x)+sl κ(t n(u))(x) = 1 Ω (x)u(x), x R 3 Û Ö Û Ö DL κ := +G κ T n : H d() X(,Ω) SL κ := G κ T d : H n() X(,Ω) Explicit expression DL κ(q,β)(x) := ( G κ)(x y) (q(y) n(y))+g κ(x y)n(y)β(y) dσ(y) SL κ(p,α)(x) := G κ(x y)p(y)+( G)(x y)α(y) dσ(y) No blow up at low frequency as the kernel G κ(x) G 0 (x) remains bounded for κ 0.
22 Jump formula and Calderón s operator Proposition : for = d, n, denotingt,c = exterior traces, and [T ] := T T,c, we have [T d] DL κ = Á, [T d] SL κ = 0, [T n] DL κ = 0, [T n] SL κ = Á. Proposition : the matrix of continuous boundary integral operators [ Td DL κ ] T d SL κ C := T n DL κ T n SL κ is a projector C 2 = C mappingh d() H n() H d() H n(). For u X(,Ω) we have u +κ 2 u = 0 in Ω (T d(u),t n(u)) Ê Ò (C). Proposition : Each of the four entries of the Calderón projector C is an invertible operator, unless κ 2 is an eigenvalue of in Ω.
23 First kind boundary integral operators In [Mitrea & al, 2016], focus was on the BIOs of the second kind T d DL κ and T n SL κ. Here, we focus on integral operators of the first kind. They admit the following variational form. ) ), = G κ(x y)[ p(y) q(x)+α(y) Ú q(x) ]dσ ( p T d SL κ α ( p T d SL κ α ), ( q β ( q β ) = and denoting p (y) = n(y) p(y), G κ(x y)[ Ú q(y)β(x)+κ 2 α(y)β(x) ]dσ ( p ) ( q ) T n DL κ, = α β G κ(x y)[ Ú q (x) Ú p (y) κ 2 q (x) p (y) ]dσ + G κ(x y)[ q (x) (n(y) α(y))+p (y) (n(x) β(x)) ]dσ + G κ(x y)α(y)β(x)n(y) n(x) dσ
24 Garding s inequality A classical tool for proving Garding s inequality for Maxwell related operators (see e.g. [Buffa & Hiptmair, 2002]) is the existence of a projector É : H 1/2 ( Ú,) H 1/2 ( Ú,) such that aaaaa É maps continuously into H 1/2 r () := {n u, u À 1 (Ω) 3 } aaaaa Ö(É) = {u H 1/2 ( Ú,), Ú (u) = 0}. Define an involutionθ 2 = Á by [ ] [ p p Θ( ) := α α ] [ 2 É(p) κ 2 Ú (p) ]. Theorem : For any κ R +, there exists a constant c(κ) > 0 and a compact operator à : H n() H n() such that Re{ (T d SL κ + à κ)u,θ(u) } c(κ) u 2 H n() u H n(). Remarks : Remarks : Analogous result holds for T n DL κ The coercivity constant depends on κ
25 Outline I.II Maxwell as vector Helmholtz equation II.I Hodge-Helmholtz potential theory III. Low frequency regime
26 Explicit expression of low frequency operators We are particularly interested in studying the operators at vanishing frequency. ( p ) ( q ) T d SL 0, = α β G 0 (x y)[ p(y) q(x)+α(y) Ú q(x)+β(x) Ú q(y) ]dσ Unfortunately this operator admits a finite dimensional but systematically non trivial kernel. Indeed we have. Proposition : We have T d SL 0 (p,α) = 0 if and only if p = 0 and ( α(y)dσ(y)/ x y ) = 0, so that Ñ Ö(T d SL 0 ) = #{ ÓÒÒ Ø ÓÑÔÓÒ ÒØ Ó } Ñ Ö(T d SL 0 ) = ½ Ø ØØ ÒÙÑ Ö Ó º Proposition : Any element(p,α) = (0,α) Ö(T d SL 0 ) satisfying αβdσ = 0 for all β Ö( ) = { locally constants } vanishesα = 0.
27 Explicit expression of low frequency operators We are particularly interested in studying the operators at vanishing frequency. ( p ) ( q ) T d SL 0, = α β G 0 (x y)[ p(y) q(x)+α(y) Ú q(x)+β(x) Ú q(y) ]dσ Unfortunately this operator admits a finite dimensional but systematically non trivial kernel. Indeed we have. Proposition : We have T d SL 0 (p,α) = 0 if and only if p = 0 and ( α(y)dσ(y)/ x y ) = 0, so that Ñ Ö(T d SL 0 ) = #{ ÓÒÒ Ø ÓÑÔÓÒ ÒØ Ó } Ñ Ö(T d SL 0 ) = ½ Ø ØØ ÒÙÑ Ö Ó º Proposition : Any element(p,α) = (0,α) Ö(T d SL 0 ) satisfying αβdσ = 0 for all β Ö( ) = { locally constants } vanishesα = 0.
28 Explicit expression of low frequency operators Proposition : We have T d SL 0 (p,α) = 0 if and only if p = 0 and ( α(y)dσ(y)/ x y ) = 0, so that Ñ Ö(T d SL 0 ) = #{ ÓÒÒ Ø ÓÑÔÓÒ ÒØ Ó } Ñ Ö(T d SL 0 ) = ½ Ø ØØ ÒÙÑ Ö Ó º Proposition : Any element(p,α) = (0,α) Ö(T d SL 0 ) satisfying αβdσ = 0 for all β Ö( ) = { locally constants } vanishesα = 0. Regularised formulation : elements of the kernel can be filtered out imposing constraints/lagrange parameters, which leads to the following saddle point problem. Find u = (p,α) H n() and µ Ö( ) such that T d SL 0 (u),v + µβ dσ = f,v v = (q,β) Hn() λα dσ = 0 λ Ö( )
29 Explicit expression of low frequency operators Things are somehow more involved for T n DL 0. Denote p := n p. The variationnal form of this operator is ( p ) ( q ) T n DL 0, = α β G 0 (x y)[ α(y)β(x)n(y) n(x) Ú p (y) Ú q (x) ]dσ(x, y) G 0 (x y)[ q (x) curl α(y)+p (y) curl β(x) ]dσ(x, y) Proposition : We havet n DL 0 (p,α) if and only if α = 0 and curl p = 0 and curl ( n(y) p(y)dσ(y)/ x y ) = 0, and we have Ñ Ö(T n DL κ) = #{ ÒÓÒ¹ ÓÙÒ Ò ÝÐ ÓÒ } Ñ Ö(T n DL κ) = ¾Ò ØØ ÒÙÑ Ö Ó º Proposition : Any element(p,α) = (p, 0) Ö(T n DL 0 ) satisfying p qdσ = 0 for all q Ö(curl ) Ö( Ú ) vanishes p = 0.
30 Conclusion Summary We have devised continuity, well-posedness and coercivity analysis for layer potentials associated to the Hodge-Helmholtz in Lipschitz domains. These can be used to reformulate Maxwell s equations and they remain bounded at low frequency. Submitted : X. Claeys and R. Hiptmair, First kind boundary integral formulation for the Hodge-Helmholtz equation, SAM ETHZ report available. Questions Discrete inf-sup condition holding uniformly at low frequency (forthcoming) Numerical experiments (work in progress with E.Demaldent CEA List) Transmission problems? Effective stabilisation strategy at low frequency? Precise connections with other existing approaches? Compuation of nonbounding cylces? Formulation remaining well conditionned for wide frequency range?
31 IABEM 2018 Symposium of the International Association for Boundary Element Methods What? International conference focused on boundary integral equations Both theory and application oriented Where? University Pierre-et-Marie Curie (Paris 6) When? June 26-28, 2018 Website ØØÔ»»ÔÖÓ Øº ÒÖ º Ö» Ѿ¼½»
32 Thank you for your attention
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 informationIntegral 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 informationELECTROMAGNETIC SCATTERING FROM PERTURBED SURFACES. Katharine Ott Advisor: Irina Mitrea Department of Mathematics University of Virginia
ELECTROMAGNETIC SCATTERING FROM PERTURBED SURFACES Katharine Ott Advisor: Irina Mitrea Department of Mathematics University of Virginia Abstract This paper is concerned with the study of scattering of
More informationFinal Exam May 4, 2016
1 Math 425 / AMCS 525 Dr. DeTurck Final Exam May 4, 2016 You may use your book and notes on this exam. Show your work in the exam book. Work only the problems that correspond to the section that you prepared.
More informationLecture 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 informationThe Gauss-Green Formula (And Elliptic Boundary Problems On Rough Domains) Joint Work with Steve Hofmann and Marius Mitrea
The Gauss-Green Formula And Elliptic Boundary Problems On Rough Domains) Joint Work with Steve Hofmann and Marius Mitrea Dirichlet Problem on open in a compact Riemannian manifold M), dimension n: ) Lu
More informationBETI for acoustic and electromagnetic scattering
BETI for acoustic and electromagnetic scattering O. Steinbach, M. Windisch Institut für Numerische Mathematik Technische Universität Graz Oberwolfach 18. Februar 2010 FWF-Project: Data-sparse Boundary
More informationTechnische Universität Graz
Technische Universität Graz Modified combined field integral equations for electromagnetic scattering O. Steinbach, M. Windisch Berichte aus dem Institut für Numerische Mathematik Bericht 2007/6 Technische
More informationCoercivity 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 informationAposteriorierrorestimatesinFEEC 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 informationFinite Element Methods for Maxwell Equations
CHAPTER 8 Finite Element Methods for Maxwell Equations The Maxwell equations comprise four first-order partial differential equations linking the fundamental electromagnetic quantities, the electric field
More informationDivergence Boundary Conditions for Vector Helmholtz Equations with Divergence Constraints
Divergence Boundary Conditions for Vector Helmholtz Equations with Divergence Constraints Urve Kangro Department of Mathematical Sciences Carnegie Mellon University Pittsburgh, PA 15213 Roy Nicolaides
More informationNumerical approximation of output functionals for Maxwell equations
Numerical approximation of output functionals for Maxwell equations Ferenc Izsák ELTE, Budapest University of Twente, Enschede 11 September 2004 MAXWELL EQUATIONS Assumption: electric field ( electromagnetic
More informationA MHD problem on unbounded domains - Coupling of FEM and BEM
A MHD problem on unbounded domains - Coupling of FEM and BEM Wiebke Lemster and Gert Lube Abstract We consider the MHD problem on R 3 = Ω Ω E, where Ω is a bounded, conducting Lipschitz domain and Ω E
More informationPartial Differential Equations
Part II Partial Differential Equations Year 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2015 Paper 4, Section II 29E Partial Differential Equations 72 (a) Show that the Cauchy problem for u(x,
More informationA 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 informationTHE INTERIOR TRANSMISSION PROBLEM FOR REGIONS WITH CAVITIES
THE INTERIOR TRANSMISSION PROBLEM FOR REGIONS WITH CAVITIES FIORALBA CAKONI, AVI COLTON, AN HOUSSEM HAAR Abstract. We consider the interior transmission problem in the case when the inhomogeneous medium
More informationTrefftz-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 informationSPECTRAL PROPERTIES OF THE LAPLACIAN ON BOUNDED DOMAINS
SPECTRAL PROPERTIES OF THE LAPLACIAN ON BOUNDED DOMAINS TSOGTGEREL GANTUMUR Abstract. After establishing discrete spectra for a large class of elliptic operators, we present some fundamental spectral properties
More informationAn introduction to the use of integral equation. for Harmonic Maxwell equation. J.C. NÉDÉLEC CMAP, École Polytechnique
An introduction to the use of integral equation for Harmonic Maxwell equation J.C. NÉDÉLEC CMAP, École Polytechnique 1. Introduction Sommaire 2. The plane waves solutions 3. Fundamental Solution and Radiation
More informationTrefftz-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 informationLaplace s Equation. Chapter Mean Value Formulas
Chapter 1 Laplace s Equation Let be an open set in R n. A function u C 2 () is called harmonic in if it satisfies Laplace s equation n (1.1) u := D ii u = 0 in. i=1 A function u C 2 () is called subharmonic
More informationMixed exterior Laplace s problem
Mixed exterior Laplace s problem Chérif Amrouche, Florian Bonzom Laboratoire de mathématiques appliquées, CNRS UMR 5142, Université de Pau et des Pays de l Adour, IPRA, Avenue de l Université, 64000 Pau
More informationWeek 6 Notes, Math 865, Tanveer
Week 6 Notes, Math 865, Tanveer. Energy Methods for Euler and Navier-Stokes Equation We will consider this week basic energy estimates. These are estimates on the L 2 spatial norms of the solution u(x,
More informationSHARP BOUNDARY TRACE INEQUALITIES. 1. Introduction
SHARP BOUNDARY TRACE INEQUALITIES GILES AUCHMUTY Abstract. This paper describes sharp inequalities for the trace of Sobolev functions on the boundary of a bounded region R N. The inequalities bound (semi-)norms
More informationScientific 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 informationFinite Elements. Colin Cotter. February 22, Colin Cotter FEM
Finite Elements February 22, 2019 In the previous sections, we introduced the concept of finite element spaces, which contain certain functions defined on a domain. Finite element spaces are examples of
More informationA DELTA-REGULARIZATION FINITE ELEMENT METHOD FOR A DOUBLE CURL PROBLEM WITH DIVERGENCE-FREE CONSTRAINT
A DELTA-REGULARIZATION FINITE ELEMENT METHOD FOR A DOUBLE CURL PROBLEM WITH DIVERGENCE-FREE CONSTRAINT HUOYUAN DUAN, SHA LI, ROGER C. E. TAN, AND WEIYING ZHENG Abstract. To deal with the divergence-free
More informationTrefftz-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 informationStandard 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 informationThe Factorization Method for a Class of Inverse Elliptic Problems
1 The Factorization Method for a Class of Inverse Elliptic Problems Andreas Kirsch Mathematisches Institut II Universität Karlsruhe (TH), Germany email: kirsch@math.uni-karlsruhe.de Version of June 20,
More informationInequalities of Babuška-Aziz and Friedrichs-Velte for differential forms
Inequalities of Babuška-Aziz and Friedrichs-Velte for differential forms Martin Costabel Abstract. For sufficiently smooth bounded plane domains, the equivalence between the inequalities of Babuška Aziz
More informationWhen 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 information44 CHAPTER 3. CAVITY SCATTERING
44 CHAPTER 3. CAVITY SCATTERING For the TE polarization case, the incident wave has the electric field parallel to the x 3 -axis, which is also the infinite axis of the aperture. Since both the incident
More informationApproximation of fluid-structure interaction problems with Lagrange multiplier
Approximation of fluid-structure interaction problems with Lagrange multiplier Daniele Boffi Dipartimento di Matematica F. Casorati, Università di Pavia http://www-dimat.unipv.it/boffi May 30, 2016 Outline
More informationTransformation of corner singularities in presence of small or large parameters
Transformation of corner singularities in presence of small or large parameters Monique Dauge IRMAR, Université de Rennes 1, FRANCE Analysis and Numerics of Acoustic and Electromagnetic Problems October
More informationMéthodes d énergie pour le potentiel de double couche: Une inégalité de Poincaré oubliée.
Méthodes d énergie pour le potentiel de double couche: Une inégalité de Poincaré oubliée. Martin Costabel IRMAR, Université de Rennes 1 2e Journée de l équipe d analyse numérique Rennes 25 oct 2007 Martin
More informationFAST INTEGRAL EQUATION METHODS FOR THE MODIFIED HELMHOLTZ EQUATION
FAST INTEGRAL EQUATION METHODS FOR THE MODIFIED HELMHOLTZ EQUATION by Bryan Quaife B.Sc., University of Calgary, 2004 M.Sc., University of Calgary, 2006 a Thesis submitted in partial fulfillment of the
More informationNumerical Solutions to Partial Differential Equations
Numerical Solutions to Partial Differential Equations Zhiping Li LMAM and School of Mathematical Sciences Peking University Sobolev Embedding Theorems Embedding Operators and the Sobolev Embedding Theorem
More informationNavier-Stokes equations in thin domains with Navier friction boundary conditions
Navier-Stokes equations in thin domains with Navier friction boundary conditions Luan Thach Hoang Department of Mathematics and Statistics, Texas Tech University www.math.umn.edu/ lhoang/ luan.hoang@ttu.edu
More informationBayesian inverse problems with Laplacian noise
Bayesian inverse problems with Laplacian noise Remo Kretschmann Faculty of Mathematics, University of Duisburg-Essen Applied Inverse Problems 2017, M27 Hangzhou, 1 June 2017 1 / 33 Outline 1 Inverse heat
More informationFinite Element Exterior Calculus. Douglas N. Arnold, University of Minnesota The 41st Woudschoten Conference 5 7 October 2016
Finite Element Exterior Calculus Douglas N. Arnold, University of Minnesota The 41st Woudschoten Conference 5 7 October 2016 The Hilbert complex framework Discretization of Hilbert complexes Finite element
More informationChapter 7: Bounded Operators in Hilbert Spaces
Chapter 7: Bounded Operators in Hilbert Spaces I-Liang Chern Department of Applied Mathematics National Chiao Tung University and Department of Mathematics National Taiwan University Fall, 2013 1 / 84
More informationarxiv: v2 [math-ph] 4 Apr 2014
The Decoupled Potential Integral Equation for Time-Harmonic Electromagnetic Scattering Felipe Vico Leslie Greengard Miguel Ferrando Zydrunas Gimbutas arxiv:144.749v2 [math-ph] 4 Apr 214 April 7, 214 Abstract
More informationVariational 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 informationAsymptotics for posterior hazards
Asymptotics for posterior hazards Pierpaolo De Blasi University of Turin 10th August 2007, BNR Workshop, Isaac Newton Intitute, Cambridge, UK Joint work with Giovanni Peccati (Université Paris VI) and
More informationOn a class of stochastic differential equations in a financial network model
1 On a class of stochastic differential equations in a financial network model Tomoyuki Ichiba Department of Statistics & Applied Probability, Center for Financial Mathematics and Actuarial Research, University
More informationTechnische 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 informationWeak Formulation of Elliptic BVP s
Weak Formulation of Elliptic BVP s There are a large number of problems of physical interest that can be formulated in the abstract setting in which the Lax-Milgram lemma is applied to an equation expressed
More informationBOUNDARY VALUE PROBLEMS ON A HALF SIERPINSKI GASKET
BOUNDARY VALUE PROBLEMS ON A HALF SIERPINSKI GASKET WEILIN LI AND ROBERT S. STRICHARTZ Abstract. We study boundary value problems for the Laplacian on a domain Ω consisting of the left half of the Sierpinski
More informationVolume and surface integral equations for electromagnetic scattering by a dielectric body
Volume and surface integral equations for electromagnetic scattering by a dielectric body M. Costabel, E. Darrigrand, and E. H. Koné IRMAR, Université de Rennes 1,Campus de Beaulieu, 35042 Rennes, FRANCE
More informationFrequency Domain Integral Equations for Acoustic and Electromagnetic Scattering Problems
Frequency Domain Integral Equations for Acoustic and Electromagnetic Scattering Problems Christophe Geuzaine, U of Liege Fernando Reitich, U of Minnesota Catalin Turc, UNC Charlotte Outline Integral equations
More informationTHE L 2 -HODGE THEORY AND REPRESENTATION ON R n
THE L 2 -HODGE THEORY AND REPRESENTATION ON R n BAISHENG YAN Abstract. We present an elementary L 2 -Hodge theory on whole R n based on the minimization principle of the calculus of variations and some
More informationIntroduction 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 informationFinite-time singularity formation for Euler vortex sheet
Finite-time singularity formation for Euler vortex sheet Daniel Coutand Maxwell Institute Heriot-Watt University Oxbridge PDE conference, 20-21 March 2017 Free surface Euler equations Picture n N x Ω Γ=
More informationIs 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 informationLECTURE # 0 BASIC NOTATIONS AND CONCEPTS IN THE THEORY OF PARTIAL DIFFERENTIAL EQUATIONS (PDES)
LECTURE # 0 BASIC NOTATIONS AND CONCEPTS IN THE THEORY OF PARTIAL DIFFERENTIAL EQUATIONS (PDES) RAYTCHO LAZAROV 1 Notations and Basic Functional Spaces Scalar function in R d, d 1 will be denoted by u,
More informationON THE EXISTENCE OF TRANSMISSION EIGENVALUES. Andreas Kirsch1
Manuscript submitted to AIMS Journals Volume 3, Number 2, May 29 Website: http://aimsciences.org pp. 1 XX ON THE EXISTENCE OF TRANSMISSION EIGENVALUES Andreas Kirsch1 University of Karlsruhe epartment
More informationPreconditioned 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 informationChapter 1 Mathematical Foundations
Computational Electromagnetics; Chapter 1 1 Chapter 1 Mathematical Foundations 1.1 Maxwell s Equations Electromagnetic phenomena can be described by the electric field E, the electric induction D, the
More informationPARTIAL DIFFERENTIAL EQUATIONS. Lecturer: D.M.A. Stuart MT 2007
PARTIAL DIFFERENTIAL EQUATIONS Lecturer: D.M.A. Stuart MT 2007 In addition to the sets of lecture notes written by previous lecturers ([1, 2]) the books [4, 7] are very good for the PDE topics in the course.
More informationThe Dirichlet problem for non-divergence parabolic equations with discontinuous in time coefficients in a wedge
The Dirichlet problem for non-divergence parabolic equations with discontinuous in time coefficients in a wedge Vladimir Kozlov (Linköping University, Sweden) 2010 joint work with A.Nazarov Lu t u a ij
More informationON THE KLEINMAN-MARTIN INTEGRAL EQUATION METHOD FOR ELECTROMAGNETIC SCATTERING BY A DIELECTRIC BODY
ON THE KLEINMAN-MARTIN INTEGRAL EQUATION METHOD FOR ELECTROMAGNETIC SCATTERING BY A DIELECTRIC BODY MARTIN COSTABEL AND FRÉDÉRIQUE LE LOUËR Abstract. The interface problem describing the scattering of
More informationMINIMAL GRAPHS PART I: EXISTENCE OF LIPSCHITZ WEAK SOLUTIONS TO THE DIRICHLET PROBLEM WITH C 2 BOUNDARY DATA
MINIMAL GRAPHS PART I: EXISTENCE OF LIPSCHITZ WEAK SOLUTIONS TO THE DIRICHLET PROBLEM WITH C 2 BOUNDARY DATA SPENCER HUGHES In these notes we prove that for any given smooth function on the boundary of
More informationLECTURE 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 informationOn a weighted total variation minimization problem
On a weighted total variation minimization problem Guillaume Carlier CEREMADE Université Paris Dauphine carlier@ceremade.dauphine.fr Myriam Comte Laboratoire Jacques-Louis Lions, Université Pierre et Marie
More informationSCHRIFTENREIHE DER FAKULTÄT FÜR MATHEMATIK. On Maxwell s and Poincaré s Constants. Dirk Pauly SM-UDE
SCHIFTENEIHE DE FAKULTÄT FÜ MATHEMATIK On Maxwell s and Poincaré s Constants by Dirk Pauly SM-UDE-772 2013 On Maxwell s and Poincaré s Constants Dirk Pauly November 11, 2013 Dedicated to Sergey Igorevich
More informationThe continuity method
The continuity method The method of continuity is used in conjunction with a priori estimates to prove the existence of suitably regular solutions to elliptic partial differential equations. One crucial
More informationIntroduction and some preliminaries
1 Partial differential equations Introduction and some preliminaries A partial differential equation (PDE) is a relationship among partial derivatives of a function (or functions) of more than one variable.
More informationRegularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian
Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian Luis Caffarelli, Sandro Salsa and Luis Silvestre October 15, 2007 Abstract We use a characterization
More informationIntroduction to the Boundary Element Method
Introduction to the Boundary Element Method Salim Meddahi University of Oviedo, Spain University of Trento, Trento April 27 - May 15, 2015 1 Syllabus The Laplace problem Potential theory: the classical
More informationA priori error analysis of the BEM with graded meshes for the electric eld integral equation on polyhedral surfaces
A priori error analysis of the BEM with graded meshes for the electric eld integral equation on polyhedral surfaces A. Bespalov S. Nicaise Abstract The Galerkin boundary element discretisations of the
More informationTraces, extensions and co-normal derivatives for elliptic systems on Lipschitz domains
Traces, extensions and co-normal derivatives for elliptic systems on Lipschitz domains Sergey E. Mikhailov Brunel University West London, Department of Mathematics, Uxbridge, UB8 3PH, UK J. Math. Analysis
More informationON MULTIPLE FREQUENCY POWER DENSITY MEASUREMENTS II. THE FULL MAXWELL S EQUATIONS. GIOVANNI S. ALBERTI University of Oxford
Report no. OxPDE-3/2 ON MULTIPLE FREQUENCY POWER DENSITY MEASUREMENTS II. THE FULL MAXWELL S EQUATIONS by GIOVANNI S. ALBERTI University of Oxford Oxford Centre for Nonlinear PDE Mathematical Institute,
More informationFundamental Solutions and Green s functions. Simulation Methods in Acoustics
Fundamental Solutions and Green s functions Simulation Methods in Acoustics Definitions Fundamental solution The solution F (x, x 0 ) of the linear PDE L {F (x, x 0 )} = δ(x x 0 ) x R d Is called the fundamental
More informationA Two-Scale Adaptive Integral Method
A Two-Scale Adaptive Integral Method Ali Yilmaz Department of Electrical & Computer Engineering University of Texas at Austin IEEE APS International Symposium USC/URSI ational Radio Science Meeting San
More informationMath The Laplacian. 1 Green s Identities, Fundamental Solution
Math. 209 The Laplacian Green s Identities, Fundamental Solution Let be a bounded open set in R n, n 2, with smooth boundary. The fact that the boundary is smooth means that at each point x the external
More informationThe oblique derivative problem for general elliptic systems in Lipschitz domains
M. MITREA The oblique derivative problem for general elliptic systems in Lipschitz domains Let M be a smooth, oriented, connected, compact, boundaryless manifold of real dimension m, and let T M and T
More informationShort-time expansions for close-to-the-money options under a Lévy jump model with stochastic volatility
Short-time expansions for close-to-the-money options under a Lévy jump model with stochastic volatility José Enrique Figueroa-López 1 1 Department of Statistics Purdue University Statistics, Jump Processes,
More informationError analysis and fast solvers for high-frequency scattering problems
Error analysis and fast solvers for high-frequency scattering problems I.G. Graham (University of Bath) Woudschoten October 2014 High freq. problem for the Helmholtz equation Given an object Ω R d, with
More informationOn the Spectrum of Volume Integral Operators in Acoustic Scattering
11 On the Spectrum of Volume Integral Operators in Acoustic Scattering M. Costabel IRMAR, Université de Rennes 1, France; martin.costabel@univ-rennes1.fr 11.1 Volume Integral Equations in Acoustic Scattering
More informationComparison Results for Semilinear Elliptic Equations in Equimeasurable Domains
Comparison Results for Semilinear Elliptic Equations in Equimeasurable Domains François HAMEL Aix-Marseille University & Institut Universitaire de France In collaboration with Emmanuel RUSS Workshop on
More informationLaplace equation. In this chapter we consider Laplace equation in d-dimensions given by. + u x2 x u xd x d. u x1 x 1
Chapter 6 Laplace equation In this chapter we consider Laplace equation in d-dimensions given by u x1 x 1 + u x2 x 2 + + u xd x d =. (6.1) We study Laplace equation in d = 2 throughout this chapter (excepting
More informationThe Convergence of Mimetic Discretization
The Convergence of Mimetic Discretization for Rough Grids James M. Hyman Los Alamos National Laboratory T-7, MS-B84 Los Alamos NM 87545 and Stanly Steinberg Department of Mathematics and Statistics University
More information1. Introduction, notation, and main results
Publ. Mat. 62 (2018), 439 473 DOI: 10.5565/PUBLMAT6221805 HOMOGENIZATION OF A PARABOLIC DIRICHLET PROBLEM BY A METHOD OF DAHLBERG Alejandro J. Castro and Martin Strömqvist Abstract: Consider the linear
More informationSobolev 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 informationBoundary 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 information6.3 Fundamental solutions in d
6.3. Fundamental solutions in d 5 6.3 Fundamental solutions in d Since Lapalce equation is invariant under translations, and rotations (see Exercise 6.4), we look for solutions to Laplace equation having
More informationA COLLOCATION METHOD FOR SOLVING THE EXTERIOR NEUMANN PROBLEM
TUDIA UNIV. BABEŞ BOLYAI, MATHEMATICA, Volume XLVIII, Number 3, eptember 2003 A COLLOCATION METHOD FOR OLVING THE EXTERIOR NEUMANN PROBLEM ANDA MICULA Dedicated to Professor Gheorghe Micula at his 60 th
More informationIn this chapter we study elliptical PDEs. That is, PDEs of the form. 2 u = lots,
Chapter 8 Elliptic PDEs In this chapter we study elliptical PDEs. That is, PDEs of the form 2 u = lots, where lots means lower-order terms (u x, u y,..., u, f). Here are some ways to think about the physical
More informationDomain Decomposition Preconditioners for Spectral Nédélec Elements in Two and Three Dimensions
Domain Decomposition Preconditioners for Spectral Nédélec Elements in Two and Three Dimensions Bernhard Hientzsch Courant Institute of Mathematical Sciences, New York University, 51 Mercer Street, New
More informationHigh-order quadratures for boundary integral equations: a tutorial
High-order quadratures for boundary integral equations: a tutorial CBMS conference on fast direct solvers 6/23/14 Alex Barnett (Dartmouth College) Slides accompanying a partly chalk talk. Certain details,
More informationu xx + u yy = 0. (5.1)
Chapter 5 Laplace Equation The following equation is called Laplace equation in two independent variables x, y: The non-homogeneous problem u xx + u yy =. (5.1) u xx + u yy = F, (5.) where F is a function
More informationNumerische Mathematik
Numer. Math. (2012) 122:61 99 DOI 10.1007/s00211-012-0456-x Numerische Mathematik C 0 elements for generalized indefinite Maxwell equations Huoyuan Duan Ping Lin Roger C. E. Tan Received: 31 July 2010
More informationNAVIER-STOKES EQUATIONS IN THIN 3D DOMAINS WITH NAVIER BOUNDARY CONDITIONS
NAVIER-STOKES EQUATIONS IN THIN 3D DOMAINS WITH NAVIER BOUNDARY CONDITIONS DRAGOŞ IFTIMIE, GENEVIÈVE RAUGEL, AND GEORGE R. SELL Abstract. We consider the Navier-Stokes equations on a thin domain of the
More informationSTOKES PROBLEM WITH SEVERAL TYPES OF BOUNDARY CONDITIONS IN AN EXTERIOR DOMAIN
Electronic Journal of Differential Equations, Vol. 2013 2013, No. 196, pp. 1 28. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu STOKES PROBLEM
More informationINSTITUTE OF MATHEMATICS THE CZECH ACADEMY OF SCIENCES. L q -solution of the Neumann, Robin and transmission problem for the scalar Oseen equation
INSTITUTE OF MATHEMATICS THE CZECH ACADEMY OF SCIENCES L q -solution of the Neumann, Robin and transmission problem for the scalar Oseen equation Dagmar Medková Preprint No. 24-2017 PRAHA 2017 L q -SOLUTION
More informationSimple Examples on Rectangular Domains
84 Chapter 5 Simple Examples on Rectangular Domains In this chapter we consider simple elliptic boundary value problems in rectangular domains in R 2 or R 3 ; our prototype example is the Poisson equation
More informationFrom 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