Abstract
We consider GMRES applied to discretisations of the high-frequency Helmholtz equation with strong trapping; recall that in this situation the problem is exponentially ill-conditioned through an increasing sequence of frequencies. Our main focus is on boundary-integral-equation formulations of the exterior Dirichlet and Neumann obstacle problems in 2- and 3-d. Under certain assumptions about the distribution of the eigenvalues of the integral operators, we prove upper bounds on how the number of GMRES iterations grows with the frequency; we then investigate numerically the sharpness (in terms of dependence on frequency) of both our bounds and various quantities entering our bounds. This paper is therefore the first comprehensive study of the frequency-dependence of the number of GMRES iterations for Helmholtz boundary-integral equations under trapping.
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Notes
It was recently shown in [28], however, that there exist star-shaped Lipschitz polyhedra such that \(A_{k}^{\prime }\) cannot be written as the sum of a coercive operator and a compact operator for any k >β0, and thus cannot be written as aI + K, where aβ β0 and K is compact, see [28, Corollary 1.6].
Note that [86] only considers the Dirichlet problem.
In [17], Ξ©+ is assumed to contain the whole ellipse E. However, inspecting the proof, we see that the result remains unchanged if E is replaced with the convex hull of the neighbourhoods of (0,Β±a2).
References
Amini, S.: On the choice of the coupling parameter in boundary integral formulations of the exterior acoustic problem. Appl. Anal. 35(1-4), 75β92 (1990)
Anderson, E., Bai, Z., Bischof, C., Blackford, S., Demmel, J., Dongarra, J., Croz, J. D. u., Greenbaum, A., Hammarling, S., McKenney, A., Sorensen, D.: LAPACK Usersβ Guide, 3rd edn. Society for Industrial and Applied Mathematics, Philadelphia (1999)
Antoine, X., Darbas, M.: Alternative integral equations for the iterative solution of acoustic scattering problems. Quart. J. Mech Appl. Math. 58(1), 107β128 (2005)
Antoine, X., Darbas, M.: Generalized combined field integral equations for the iterative solution of the three-dimensional Helmholtz equation. ESAIM: Math. Modelli. Numer. Anal. (M2AN) 41(1), 147 (2007)
Atkinson, K.: Convergence rates for approximate eigenvalues of compact integral operators. SIAM J. Numer. Anal. 12(2), 213β222 (1975)
Atkinson, K.E.: The numerical solution of the eigenvalue problem for compact integral operators. Trans. Am. Math. Soc. 129(3), 458β465 (1967)
Atkinson, K. E.: The numerical solution of integral equations of the second kind cambridge monographs on applied and computational mathematics (1997)
AvakumoviΔ, V.G.: ΓΌber die Eigenfunktionen auf geschlossenen Riemannschen Mannigfaltigkeiten. Math. Z. 65, 327β344 (1956)
Balay, S., Abhyankar, S., Adams, M.F., Brown, J., Brune, P., Buschelman, K., Dalcin, L., Eijkhout, V., Gropp, W.D., Karpeyev, D., Kaushik, D., Knepley, M.G., May, D.A., Curfman McInnes, L., Tran Mills, R., Munson, T., Rupp, K., Sanan, P., Smith, B. F., Zampini, S., Zhang, H., manual, H. Zhang.: PETSC users Technical Report ANL-95/11 - Revision 3.11. Argonne National Laboratory (2019)
Balay, S., Gropp, W.D., Curfman McInnes, L., Smith, B.F.: Efficient management of parallelism in object oriented numerical software libraries. In: Arge, E., Bruaset, A.M., Langtangen, H.P. (eds.) Modern Software Tools in Scientific Computing, pp 163β202. BirkhΓ€user Press (1997)
Bao, G., Sun, W.: A fast algorithm for the electromagnetic scattering from a large cavity. SIAM J. Sci. Comput. 27(2), 553β574 (2005)
Barnett, A.: MPSPack tutorial. https://github.com/ahbarnett/mpspack/blob/master/doc/tutorial.pdf (2006)
Barnett, A., Hassell, A.: Fast computation of high-requency Dirichlet eigenmodes via spectral flow of the interior Neumann-to-Dirichlet map. Commun. Pure Appl. Math. 67(3), 351β407 (2014)
Barnett, A.H., Betcke, T., Quantum mushroom billiards. Chaos: Interdiscip. J. Nonlinear Sci. 17(4), 043125 (2007)
Baskin, D., Spence, E.A., Wunsch, J.: Sharp high-frequency estimates for the Helmholtz equation and applications to boundary integral equations. SIAM J. Math. Anal. 48(1), 229β267 (2016)
Beckermann, B., Goreinov, S.A., Tyrtyshnikov, E.E.: Some remarks on the Elman, estimate for GMRES. SIAM J. Matrix Anal. Appl. 27(3), 772β778 (2005)
Betcke, T., Chandler-Wilde, S.N., Graham, I.G., Langdon, S., Lindner, M.: Condition number estimates for combined potential boundary integral operators in acoustics and their boundary element discretisation. Numer. Methods Partial Differ. Equ. 27(1), 31β69 (2011)
Betcke, T., Phillips, J., Spence, E.A.: Spectral decompositions and non-normality of boundary integral operators in acoustic scattering. IMA J. Num. Anal. 34(2), 700β731 (2014)
Betcke, T., Spence, E.A.: Numerical estimation of coercivity constants for boundary integral operators in acoustic scattering. SIAM J. Numer. Anal. 49(4), 1572β1601 (2011)
Boubendir, Y., Turc, C.: Wave-number estimates for regularized combined field boundary integral operators in acoustic scattering problems with Neumann boundary conditions. IMA J. Numer. Anal. 33(4), 1176β1225 (2013)
Bruno, O., Elling, T., Turc, C.: Regularized integral equations and fast high-order solvers for sound-hard acoustic scattering problems. Int. J. Numer. Methods Eng. 91(10), 1045β1072 (2012)
Burq, N.: DΓ©croissance des ondes absence de de lβΓ©nergie locale de lβΓ©quation pour le problΓ¨me extΓ©rieur et absence de resonance au voisinage du rΓ©el. Acta Math. 180, 1β29 (1998)
Campbell, S.L., Ipsen, I.C.F., Kelley, C.T., Meyer, C.D.: GMRES And the minimal polynomial. BIT Numer. Math. 36(4), 664β675 (1996)
Canzani, Y., Galkowski, J.: Weyl remainders: an application of geodesic beams. arXiv:2010.03969 (2020)
Cardoso, F., Popov, G.: Quasimodes with exponentially small errors associated with elliptic periodic rays. Asymptot. Anal. 30(3, 4), 217β247 (2002)
Chandler-Wilde, S.N., Graham, I.G., Langdon, S., Lindner, M.: Condition number estimates for combined potential boundary integral operators in acoustic scattering. J. Integral Equ. Appl. 21(2), 229β279 (2009)
Chandler-Wilde, S.N., Graham, I.G., Langdon, S., Spence, E.A.: Numerical-asymptotic boundary integral methods in high-frequency acoustic scattering. Acta Numer. 21(1), 89β305 (2012)
Chandler-Wilde, S.N., Spence, E.A.: Coercivity, essential norms, and the Galerkin method for second-kind integral equations on polyhedral and Lipschitz domains. arXiv:2105.11383 (2021)
Chandler-Wilde, S.N., Spence, E.A., Gibbs, A., Smyshlyaev, V.P.: High-frequency bounds for the Helmholtz equation under parabolic trapping and applications in numerical analysis. SIAM J. Math. Anal. 52(1), 845β893 (2020)
Colin de VerdiΓ¨re, Y.: On the remainder in the Weyl formula for the Euclidean disk. SΓ©m. ThΓ©or. Spectrale GΓ©om. 29, 1β13 (2010)
Darbas, M., Darrigrand, E., Lafranche, Y.: Combining analytic preconditioner and fast multipole method for the 3-D Helmholtz equation. J. Comput. Phys. 236, 289β316 (2013)
Du, K.: GMRES With adaptively deflated restarting and its performance on an electromagnetic cavity problem. Appl. Numer. Math. 61(9), 977β988 (2011)
Duistermaat, J.J., Guillemin, V.W.: The spectrum of positive elliptic operators and periodic bicharacteristics. Invent. Math. 29(1), 39β79 (1975)
Dyatlov, S., Zworski, M.: Mathematical theory of scattering resonances. AMS (2019)
Elman, H.C., Silvester, D.J., Wathen, A.J.: Performance and analysis of saddle point preconditioners for the discrete steady-state Navier-Stokes equations. Numer. Math. 90(4), 665β688 (2002)
Embree, M.: How descriptive are gmres convergence bounds? Technical report, Oxford University Computing Laboratory (1999)
Erlangga, Y.A.: Advances in iterative methods and preconditioners for the Helmholtz equation. Arch. Comput. Methods Eng. 15(1), 37β66 (2008)
Ernst, O.G., Gander, M.J.: Why it is difficult to solve Helmholtz problems with classical iterative methods. In: Graham, I. G., Hou, T.Y., Lakkis, O., Scheichl, R. (eds.) Numerical Analysis of Multiscale Problems, volume 83 of Lecture Notes in Computational Science and Engineering, pp 325β363. Springer (2012)
Fabes, E.B., Jodeit, M., Riviere, N.M.: Potential techniques for boundary value problems on C1 domains. Acta Math. 141(1), 165β186 (178)
Feynman, R.P., Leighton, R.B., Sands, M.: The Feynman lectures on physics, vol. 1. Addison-Wesley (1964)
Fricker, F.: EinfΓΌhrung in die Gitterpunktlehre, volume 73 of LehrbΓΌcher und Monographien aus dem Gebiete der Exakten Wissenschaften (LMW). Mathematische Reihe [Textbooks and Monographs in the Exact Sciences. Mathematical Series]. BirkhΓ€user Verlag, Basel-Boston Mass (1982)
Galkowski, J.: Distribution of resonances in scattering by thin barriers. Mem. Amer. Math. Soc. 259(1248), ix+β152 (2019)
Galkowski, J., Lafontaine, D., Spence, E.A.: Local absorbing boundary conditions on fixed domains give order-one errors for high-frequency waves. arXiv:2101.02154 (2021)
Galkowski, J., Marchand, P., Spence, E.A.: Eigenvalues of the truncated Helmholtz solution operator under strong trapping. SIAM J. Math. Anal., to appear (2021)
Galkowski, J., Marchand, P., Spence, E.A.: High-frequency estimates on boundary integral operators for the Helmholtz exterior Neumann problem. arXiv:2109.06017 (2021)
Galkowski, J., MΓΌller, E.H., Spence, E.A.: Wavenumber-explicit analysis for the Helmholtz h-BEM: error estimates and iteration counts for the Dirichlet problem. Numer. Math. 142(2), 329β357 (2019)
Galkowski, J., Smith, H.F.: Restriction bounds for the free resolvent and resonances in lossy scattering. Internat. Math. Res. Notices 16, 7473β7509 (2015)
Galkowski, J., Spence, E.A.: Wavenumber-explicit regularity estimates on the acoustic single-and double-layer operators. Integr. Equat. Oper. Th. 91(6) (2019)
Galkowski, J., Toth, J.A.: Pointwise bounds for joint eigenfunctions of quantum completely integrable systems. Commun. Math. Phys. 375 (2), 915β947 (2020)
Gander, M.J., Zhang, H.: A class of iterative solvers for the Helmholtz equation: factorizations, sweeping preconditioners, source transfer, single layer potentials, polarized traces, and optimized Schwarz methods. SIAM Rev. 61(1), 3β76 (2019)
Giraud, L., Gratton, S., Pinel, X., Vasseur, X.: Flexible GMRES with deflated restarting. SIAM J. Sci. Comput. 32(4), 1858β1878 (2010)
Gmati, N., Philippe, B.: Comments on the GMRES convergence for preconditioned systems. In: Large-Scale Scientific Computing, pp. 40β51. Springer (2007)
Graham, I.G., LΓΆhndorf, M., Melenk, J.M., Spence, E.A.: When is the error in the h-BEM for solving the Helmholtz equation bounded independently of k?. BIT Numer Math. 55(1), 171β214 (2015)
Graham, I.G., Spence, E.A., Zou, J.: Domain decomposition with local impedance conditions for the helmholtz equation with absorption. SIAM J. Numer. Anal. 58(5), 2515β2543 (2020)
Han, X., Tacy, M.: Sharp norm estimates of layer potentials and operators at high frequency. J. Funct. Anal. 269, 2890β2926. With an appendix by Jeffrey Galkowski (2015)
Hardy, G.H.: On the expression of a number as the sum of two squares. Quart. J. Math. 46, 263β283 (1915)
Hecht, F.: New development in FreeFem++. J. Numer. Math. 20(3-4), 251β266 (2012)
Hiptmair, R.: Operator preconditioning. Comput. Math. Appl. 52(5), 699β706 (2006)
HΓΆrmander, L.: The spectral function of an elliptic operator. Acta Math. 121, 193β218 (1968)
HΓΆrmander, L.: The analysis of linear partial differential operators IV: Fourier Integral Operators. Springer (1985)
IvriΔ±Μ, V.J.: The second term of the spectral asymptotics for a Laplace-Beltrami operator on manifolds with boundary. Funktsional. Anal. Prilozhen. 14(2), 25β34 (1980)
Jennings, A.: Influence of the eigenvalue spectrum on the convergence rate of the conjugate gradient method. IMA J. Appl. Math. 20(1), 61β72 (1977)
Kirby, R.C.: From functional analysis to iterative methods. SIAM Rev. 52(2), 269β293 (2010)
Koch, T., Liesen, J.: The conformal βbratwurstβ maps and associated Faber polynomials. Numer. Math. 86(1), 173β191 (2000)
Kress, R.: Minimizing the condition number of boundary integral operators in acoustic and electromagnetic scattering. Q. J. Mech. Appl. Math. 38 (2), 323 (1985)
Kress, R., Spassov, W.T.: On the condition number of boundary integral operators in acoustic and electromagnetic scattering. Numer. Math. 42, 77β95 (1983)
Lafontaine, D., Spence, E.A., frequencies, J. Wunsch.: For most strong trapping has a weak effect in frequency-domain scattering. Comm. Pure Appl Math (2020)
Lai, J., Ambikasaran, S., Greengard, L.F.: A fast direct solver for high frequency scattering from a large cavity in two dimensions. SIAM J. Sci. Comput. 36(6), B887βB903 (2014)
Lai, J., Greengard, L., OβNeil, M.: Robust integral formulations for electromagnetic scattering from three-dimensional cavities. J. Comput. Phys. 345, 1β16 (2017)
PhαΊ‘m, P.T.: Meilleures estimations asymptotiques des restes de la fonction spectrale et des valeurs propres relatifs au Laplacien. Math. Scand. 48 (1), 5β38 (1981)
Levitan, B.M.: On the asymptotic behavior of the spectral function of a self-adjoint differential equation of the second order. Izvestiya Akad. Nauk SSSR Ser. Mat. 16, 325β352 (1952)
Li, X.S., Demmel, J.W.: superLU_DIST A scalable distributed-memory sparse direct solver for unsymmetric linear systems. ACM Trans. Math. Softw. 29(2), 110β140 (2003)
Li, Y., Wu, H.: FEM and CIP-FEM for Helmholtz Equation with High Wave Number and Perfectly Matched Layer Truncation. SIAM J. Numer. Anal. 57(1), 96β126 (2019)
Liesen, J., Tichα»³, P.: Convergence analysis of Krylov subspace methods. GAMM-Mitteilungen 27(2), 153β173 (2004)
Liu, X., Xi, Y., Saad, Y., de Hoop, M.V.: Solving the three-dimensional high-frequency Helmholtz equation using contour integration and polynomial preconditioning. SIAM J. Matrix Anal. Appl. 41(1), 58β82 (2020)
LΓΆhndorf, M., Melenk, J.M.: Wavenumber-explicit hp-BEM for high frequency scattering. SIAM J. Numer. Anal. 49(6), 2340β2363 (2011)
Lynch, P.: Integrable elliptic billiards and ballyards. Eur. J. Phys. 41(1), 015005 (2019)
Marburg, S.: Six boundary elements per wavelength: is that enough. J. Comp. Acous. 10(01), 25β51 (2002)
Mathieu, Γ.: MΓ©moire sur le mouvement vibratoire dβune membrane de forme elliptique. J. Math. Pures Appl. 13, 137β203 (1968)
Mclachlan, N.W.: Theory and application of Mathieu functions (1951)
Meurant, G.: Estimates of the norm of the error in solving linear systems with FOM and GMRES. SIAM J. Sci. Comput. 33(5), 2686β2705 (2011)
Meurant, G., Tebbens, J.D.: The role eigenvalues play in forming GMRES residual norms with non-normal matrices. Numer. Algorithm. 68(1), 143β165 (2015)
Meurant, G., Tebbens, J.D.: Krylov methods for nonsymmetric linear systems from theory to computations. Springer Nature (2020)
Morgan, R.B.: A restarted GMRES method augmented with eigenvectors. SIAM J. Matrix Anal. Appl. 16(4), 1154β1171 (1995)
Morgan, R.B.: GMRES With deflated restarting. SIAM J. Sci. Comput. 24(1), 20β37 (2002)
Neves, A.G.M.: Eigenmodes and eigenfrequencies of vibrating elliptic membranes: a Klein oscillation theorem and numerical calculations. Commun. Pure Appl. Anal. 9(3), 611β624 (2010)
Nguyen, B.-T., Grebenkov, D.S.: Localization of laplacian eigenfunctions in circular, spherical, and elliptical domains. SIAM J. Appl. Math. 73 (2), 780β803 (2013)
NIST. Digital library of mathematical functions. Digital library of mathematical functions. http://dlmf.nist.gov/ (2021)
Petkov, V., Zworski, M.: BreitβWigner approximation and the distribution of resonances. Commun. Math. Phys. 204(2), 329β351 (1999)
PrΓΆssdorf, S.: Linear integral equations. In: Analysis IV, volume 27 of Encyclopaedia of Mathematical Sciences, pp. 1β125. Springer (1991)
Rjasanow, S., Steinbach, O.: The fast solution of boundary integral equations. Springer Science & Business Media (2007)
Saad, Y.: Iterative methods for sparse linear systems. SIAM Philadelphia (2003)
Saad, Y.: Numerical Methods for Large Eigenvalue Problems, 2nd edn. SIAM (2011)
Saad, Y., Schultz, M.H.: GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 7(3), 856β869 (1986)
Sauter, S.A., Schwab, C.: Boundary element methods. Springer, Berlin (2011)
Seeley, R.: A sharp asymptotic remainder estimate for the eigenvalues of the Laplacian in a domain of r3. Adv. Math. 29(2), 244β269 (1978)
Spence, A.: On the convergence of the NystrΓΆm method for the integral equation eigenvalue problem. Numer. Math. 25(1), 57β66 (1975)
Spence, A., Thomas, K.S.: On superconvergence properties of Galerkinβs method for compact operator equations. IMA J. Numer. Anal. 3(3), 253β271 (1983)
Spence, E.A., Kamotski, I.V., Smyshlyaev, V.P.: Coercivity of combined boundary integral equations in high frequency scattering. Comm. Pure Appl. Math 68, 1587β1639 (2015)
Stefanov, P.: Quasimodes and resonances: sharp lower bounds. Duke Math. J. 99(1), 75β92 (1999)
Stefanov, P.: Resonances near the real axis imply existence of quasimodes. Compt. Rend. lβAcad. Sci.-Ser. I-Math. 330(2), 105β108 (2000)
Stefanov, P., Vodev, G.: Distribution of resonances for the Neumann problem in linear elasticity outside a strictly convex body. Duke Math. J. 78 (3), 677β714 (1995)
Stefanov, P., Vodev, G.: Neumann resonances in linear elasticity for an arbitrary body. Commun. Math. Phys. 176(3), 645β659 (1996)
Steinbach, O.: Numerical approximation methods for elliptic boundary value problems: finite and boundary elements. Springer, New York (2008)
Steinbach, O., Wendland, W.L.: The construction of some efficient preconditioners in the boundary element method. Adv. Comput. Math. 9(1), 191β216 (1998)
Tang, S.H., Zworski, M.: From quasimodes to resonances. Math. Res. Lett. 5, 261β272 (1998)
Titley-Peloquin, D., Pestana, J., Wathen, A.J.: GMRES convergence bounds that depend on the right-hand-side vector. IMA J. Numer. Anal. 34, 462β479 (2014)
Trefethen, L.N., Bau, IIID.: Numerical linear algebra, vol. 50. SIAM (1997)
Trefethen, L.N., Embree, M.: Spectra and pseudospectra. Princeton University Press, Princeton (2005)
TΓΌreci, H.E., Schwefel, H.G.L.: An efficient Fredholm method for the calculation of highly excited states of billiards. J. Phys. A Math. Theor. 40(46), 13869 (2007)
Vasiliev, D.G., Safarov, Yu. G.: The asymptotic distribution of eigenvalues of differential operators. In: Spectral theory of operators (Novgorod, 1989), volume 150 of Amer. Math. Soc. Transl. Ser. 2, pp 55β110. Amer. Math. Soc., Providence (1992)
Vico, F., Greengard, L., Gimbutas, Z.: Boundary integral equation analysis on the sphere. Numer. Math. 128(3), 463β487 (2014)
Vodev, G.: On the exponential bound of the cutoff resolvent. Serdica Math. J. 26(1), 49pβ58p (2000)
Wang, Y., Du, K., Sun, W.: Preconditioning iterative algorithm for the electromagnetic scattering from a large cavity. Numer. Linear Algebra Appl. 16(5), 345β363 (2009)
Wilson, H.B., Scharstein, R.W.: Computing elliptic membrane high frequencies by Mathieu and Galerkin methods. J. Eng. Math. 57(1), 41β55 (2006)
Zworski, M.: Semiclassical analysis. American Mathematical Society, Providence (2012)
Acknowledgements
EAS gratefully acknowledges discussions with Alex Barnett (Flatiron Institute) that started his interest in eigenvalues of discretisations of the Helmholtz equation under strong trapping. In addition, all the authors thank Barnett for giving them insightful comments on an earlier version of this paper. PM thanks Pierre Jolivet (Institut de Recherche en Informatique de Toulouse, CNRS) and Pierre-Henri Tournier (Sorbonne UniversitΓ©, CNRS) for their help with the software FreeFEM. The authors thank the referees for their careful reading of the paper and numerous suggestions for improvement. This research made use of the Balena High Performance Computing (HPC) Service at the University of Bath. PM and EAS were supported by EPSRC grant EP/R005591/1.
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Appendices
Appendix A: Definitions of layer potentials and boundary-integral operators
The single-layer and double-layer potentials, \(\mathcal {S}_{k}\) and \(\mathcal {D}_{k}\) respectively, are defined for Ο β L1(Ξ) by
where the fundamental solution Ξ¦k is defined by
where H0(1) is the Hankel function of the first kind and order zero. If u is the solution to the scattering problem (1.1), then Greenβs second identity implies that
see, e.g. [27, Theorem 2.43].
The single-layer, adjoint-double-layer, double-layer, and hypersingular operators are defined for Ο β L2(Ξ) and Ο β H1(Ξ) by
for x βΞ. When Ξ is Lipschitz, the integrals defining Dk and \(D_{k}^{\prime }\) must be understood as Cauchy principal values (see, e.g. [27, Equation 2.33]), and the integral defining Hk is understood as a non-tangential limit (see, e.g. [27, Equation 2.36]), but we do not need the details of these definitions in this paper.
Appendix B: Bounds on the Galerkin matrix in terms of the continuous operator
Lemma 1
Let Vn β L2(Ξ) be a finite-dimensional space with real basis \(\{\phi _{j}\}_{j=1}^{n}\). Let M be defined by (1.12). Given \(A:{{L^2({\varGamma })}\rightarrow {L^2({\varGamma })}}\), let A be defined by the first equation in (1.9) (with \(A^{\prime }_{k}\) replaced by A). Let \(P_{h}:{L^2({\varGamma })}\rightarrow V_{h}\) be the orthogonal projection, and let
-
(i)
$$ (\text{cond}(\mathbf{M}))^{-1/2} \left\|\widetilde{A}\right\|_{{{L^2({\varGamma})}\rightarrow {L^2({\varGamma})}}} \leq \left\|\mathbf{M}^{-1} \mathbf{A}\right\|_2 \leq (\text{cond}(\mathbf{M}))^{1/2} \left\|\widetilde{A} \right\|_{{{L^2({\varGamma})}\rightarrow {L^2({\varGamma})}}} $$(B.1)
where \(\text {cond}(\mathbf {M}):= \|\mathbf {M}\|_{2} \|\mathbf {M}^{-1}\|_{2}\), and if (Mββ1A)ββ1 exists, then
$$ (\text{cond}(\mathbf{M}))^{-1/2}\left\|\widetilde{A}^{-1}\right\|_{{{L^2({\varGamma})}\rightarrow {L^2({\varGamma})}}}\leq \left\|(\mathbf{M}^{-1}\mathbf{A})^{-1}\right\|_2 \leq (\text{cond}(\mathbf{M}))^{1/2} \big\|\widetilde{A}^{-1}\big\|_{{{L^2({\varGamma})}\rightarrow {L^2({\varGamma})}}}. $$(B.2) -
(ii)
If \(P_{h} \phi \rightarrow \phi \) as \(h\rightarrow 0\) for all Ο β L2(Ξ), then
$$ \big\|\widetilde{A}\big\|_{{{L^2({\varGamma})}\rightarrow {L^2({\varGamma})}}} \rightarrow \left\|A\right\|_{{{L^2({\varGamma})} \rightarrow {L^2({\varGamma})}}} \quad\text{ as } h\rightarrow 0; $$(B.3)if, in addition, A = aI + K, where aβ β0 and K is compact, then
$$ \big\|\widetilde{A}^{-1}\big\|_{{{L^2({\varGamma})}\rightarrow {L^2({\varGamma})}}} \rightarrow \big\|A^{-1}\big\|_{{{L^2({\varGamma})} \rightarrow {L^2({\varGamma})}}} \quad\text{ as } h\rightarrow 0. $$(B.4)
Remark 1
For standard BEM spaces, cond(M) is bounded independently of h, see [95, Theorem 4.4.7 and Remark 4.5.3] and [104, Corollary 10.6].
Remark 2
If Ξ is C1, then both \(A_{k}^{\prime }\) and Bk,reg can be written as aI + K, where aβ β0 and K is compact. For \(A_{k}^{\prime }\) this follows since \(S_{k}:{L^2({\varGamma })}\rightarrow H^1({\varGamma })\) when Ξ is Lipschitz (see, e.g. [27, Theorem 2.17]) and \(D_{k}^{\prime }\) is compact on L2(Ξ) when Ξ is C1 by [39, Theorem 1.3].Footnote 1 For Bk,reg, the CalderΓ³n relations (see, e.g. [27, Equation 2.56]) imply that
By bounds on the fundamental solution Ξ¦k appearing in, e.g. [27, Equation 2.25], the kernel of the integral operator Hk β Hik is weakly singular, and thus the operator is compact on L2(Ξ) by, e.g. the combination of [90, Part 3 of the theorem on Page 49] and Youngβs inequality.
Proof Proof of Lemma 5
Part (ii) is proved in [17], with (B.3) proved in [17, Equation 3.3] and (B.4) proved in [17, Text below Equation 3.3]. We note that similar results are contained in [63, Β§2.4].
For Part (i), let \(\{\psi _{j}\}_{j=1}^{n}\) be an real orthonormal basis of Vn so that
Let the matrices B and D be defined by
i.e. B is the Galerkin matrix with respect to the basis \(\{\psi _{j}\}_{j=1}^{n}\), and D is a change of basis matrix from \(\{\psi _{j}\}_{j=1}^{n}\) to \(\{\phi _{j}\}_{j=1}^{n}\). Since \(\{\psi _{j}\}_{j=1}^{n}\) is orthonormal,
(provided that \(\widetilde {A}^{-1}\) exists) by, e.g. [17, Equation 3.2].
The definitions of B and D, combined with (B.6), imply that M = DDT and A = DBDT so that
Since M is symmetric positive definite, the definition \(\|\mathbf {D}\|_{2} = \sqrt {\lambda _{\max \limits }(\mathbf {D}^{T} \mathbf {D})}\) implies that \(\|\mathbf {D}\|_{2} = \|\mathbf {D}^{T}\|_{2}= \|\mathbf {M}\|_{2}^{1/2}\) and similarly \(\|\mathbf {D}^{-1}\|_{2} = \|\mathbf {D}^{-T}\|_{2}= \|\mathbf {M}^{-1}\|_{2}^{1/2}\) The results (B.1) and (B.2) then follow from combining these results about norms with (B.7) and (B.8). β‘
Remark 3 (Bounds on norms without preconditioning by M ββ1.)
The proof of Lemma 5 also implies that
so that
see, e.g. [7, Equation 3.6.166].
Appendix C: Numerical experiments about F1 and F4
Recall from Section 1.2 the features F1 and F4:
-
F1 When the incoming plane wave enters the cavity, one needs a larger number of points per wavelength for accuracy of the Galerkin solutions than when the wave does not enter the cavity.
-
F4 The GMRES residual being small does not necessarily mean that the error is small, and the relative sizes of the residual and error depend on both k and the direction of the plane wave.
Numerical experiments demonstrating F1
These experiments involve the Galerkin solutions on four different meshes with Ξ©β the small cavity. To create these meshes we start with a mesh with 10 points by wavelength, and then we refine splitting the segments in two, three, and six. The Galerkin solutions are computed using LU factorisations computed by SuperLU [72]. We use the Galerkin solution on the finest mesh as a proxy for the true solution (with this mesh denoted by Ξref), and let the three other Galerkin solutions be uh0, uh1, uh2 (with h0 > h1 > h2). Because all the meshes are obtained by refinement from the same mesh, we can interpolate a finite element function from one mesh to the other, and we let Iref denote interpolation to the reference mesh. The relative L2 Galerkin errors are defined as β₯Iref(uhi) β urefβ₯L2(Ξref)/β₯urefβ₯L2(Ξref) for i =β0, 1, 2.
FigureΒ 23 plots these errors for \(k= k_{m,0}^{e}\) for k β (50, 150) and for two different choices of the plane-wave direction: \(a=(\cos \limits \theta , \sin \limits \theta )\) with π =β4Ο/10 (the plane wave is almost vertical and enters the cavity) and π = Ο (the plane wave is horizontal, travelling from the right, and thus does not enter the cavity). In this figure, we see exactly the feature F1.
Numerical experiments demonstrating F4
These experiments work only on the mesh with 10 points by wavelength used in the experiments for F1. We compute the error between the solutions of the Galerkin equations computed by (i) SuperLU [72], denoted by \(u_{h_{0}}\) as above, and (ii) GMRES with a relative tolerance of 1 Γ 10ββ6. We then normalise this difference by β₯uh0β₯L2(Ξ). This relative L2 GMRES error is plotted in Fig.Β 24, both for π =β4Ο/10 and Ο, and we see the dependence on angle as stated in F4.
Appendix D: Discussion of the results of [107] on how the GMRES residual depends on the right-hand-side vector (relevant for F3(c))
We consider solving the linear system Bx = b with GMRES, as described in Section 1.1.7. We assume B is diagonalisable, so that
where the columns of V, denoted by \((\mathbf {v}_{1}, \mathbf {v}_{2}, \dots , \mathbf {v}_{n})\), are the right eigenvectors of B corresponding to the eigenvalues \(\lambda _{1}, \lambda _{2}, \dots , \lambda _{n}\), respectively.
We expand b as a linear combination of right eigenvectors with corresponding components Ξ²j, i.e.
and let \(\boldsymbol {\upbeta }^{\prime }:= \boldsymbol {\upbeta }/\|\mathbf {b}\|_{2}\). The result [107, Theorem 2.2] states that
i.e. the relative residual is bounded above by β₯Vβ₯2 times the residual of a polynomial least-squares approximation problem on the spectrum of B, weighted by the scaled components \({\upbeta }^{\prime }_{j}\).
We now consider the special case that the spectrum of B has a single outlier near zero, plus a cluster bounded well-away from zero; i.e. |Ξ»1|ββ0 and, for j =β2,β¦, n, Ξ»j is such that |Ξ»j β c| < Ο, with Ο βͺ|c|. In this case, the bound (D.2) becomes
In its initial stages, GMRES tries to construct a polynomial that is one at zero and is very small at Ξ»1 before dealing with the values p(Ξ»j) for the eigenvalues in the cluster. This feature of GMRES has often been remarked on, see for example, [82, Discussion after Theorem 1] and [35, Β§4 and Figure 1]. Now, if b is varied such that the relative coefficient \({\upbeta }^{\prime }_{1}\) increases, then the weight on p(Ξ»1) increases, making it harder for GMRES to make the term \(|{\upbeta }^{\prime }_{1}|^{2}|p(\lambda _{1})|^{2}\) very small. Therefore, in this special scenario of a single outlier near zero, one would expect the number of GMRES iterations to depend significantly on the size of \({\upbeta }^{\prime }_{1}\), with the number of iterations increasing as \(|{\upbeta }^{\prime }_{1}|\) increases (and decreasing if \(|{\upbeta }^{\prime }_{1}|\) decreases).
Though the argument above is heuristic and applies to a very simple situation, it illustrates that the size of the component of the right-hand-side vector in the direction of an eigenvector with corresponding eigenvalue very close to zero is likely to significantly influence the bound (D.2) on the residual in GMRES.
Appendix E: Eigenvalues and eigenfunctions of the Laplacian in an ellipse in terms of Mathieu functions
The eigenvalue problem for the Dirichlet/Neumann Laplacian in the ellipse E (1.2) is
It is customary to call {(x1, 0) : |x1|β€ a1} the major axis, {(0, x2) : |x2|β€ a2} the minor axis, π =β1 β a2 2 a12 the eccentricity, and a := a1 2 β a2 2 the linear eccentricity.
We use the following change of variables, introduced in [79],
so that
where ΞΌ0 := cosh ββ1(a1/a) = sinh ββ1(a2/a). (Note that we have used the same notation as in [17, Appendix A] for variable names etc.)
Substituting u(x1, x2) = M(ΞΌ)N(Ξ½) into (E.1), we find
where Ξ± is the separation constant and
Since (E.3) is symmetric in Ξ½, if N(Ξ½) is solution of (E.3), then so is N(βΞ½); we therefore restrict attention to solutions of (E.3) that are even or odd.
We therefore seek solutions of (E.3) and (E.4), with N even or odd, satisfying
to ensure periodicity in Ξ½, and
to ensure the zero Dirichlet/Neumann boundary condition on βE. Furthermore, to obtain a well-defined solution at ΞΌ =β0, [80] shows that we also need M to satisfy
In analogy with polar coordinates, the boundary value problem for N (E.3), (E.6) is called the angular problem, while the boundary value problem for M defined by (E.4), (E.7), and (E.8) is called the radial problem. An eigenmode u(x1, x2) = M(ΞΌ)N(Ξ½) with N is even is called an even eigenmode, and one with N odd is called an odd eigenmode.
The multiparametric spectral problems defined by (E.3), (E.6), (E.4), and (E.8a) for even modes, and (E.3), (E.6), (E.4), and (E.8b) for odd modes are well-defined by [86]Footnote 2. Indeed,
-
for (m, n) β{0, 1, 2,β¦}2, there exists a unique pair (Ξ±m, ne, qm, ne) β β Γ (0, β) such that (E.3), (E.6), (E.4), and (E.8a) have non trivial solutions Mm, ne(ΞΌ) and Nm, ne(Ξ½) with respectively m zeros in (0, ΞΌ0) and n zeros in [0, Ο),
-
for (m, n) β{0, 1, 2,β¦}Γ{1, 2,β¦}, there exists a unique pair (Ξ±m, no, qm, no) β β Γ (0, β) such that (E.3), (E.6), (E.4), and (E.8b) have a non trivial solutions Mm, no(ΞΌ) and Nm, no(Ξ½) with respectively m zeros in (0, ΞΌ0) and n zeros in [0, Ο),
Recall that q and k are related by (E.5). The frequencies associated with qm, ne and qm, no are denoted by km, ne and km, no, respectively, and the associated eigenfunctions are denoted by um, ne and um, no. By [86, Equations (21) and (22)], km, ne and km, no both increase with m and n, which is consistent with the fact the only accumulation point in the spectrum of the Laplacian is infinity.
References for the proof of Theorem 1.2
The result follows by combining the following three ingredients.
-
(i)
The results of [17, Equation A.16] and [87, Theorem 3.1] that the eigenfunctions associated with \(k_{m,n}^{e/0}\) exponentially localise about the minor axis as \(m\rightarrow \infty \) for fixed n.
-
(ii)
The arguments in [17, Proof of Theorem 2.8] that construct quasimodes of the exterior Dirichlet problem from Dirichlet eigenfunctions of the ellipse that are exponentially localised (note that these arguments also apply to the Neumann problem).Footnote 3
Regarding (i): for Ξ½0 β (0, Ο/2), let
and
i.e. \(E_{\nu _{0}}\) corresponds to the βwingsβ of the ellipse, away from the minor axis, and \(\rho _{\nu _{0}}^{e/o}(m,n)\) measures the mass of \(u^{e/o}_{m,n}\) in these regions. By [17, Equation (A.16)], there exists Ke(Ξ½0) >β0 such that ΟΞ½0e(m, 0) β² eβkm,0e for any km,0e > Ke(Ξ½0). By [87, Theorem 3.1], for n fixed, there exists Kne/o(Ξ½0) >β0 such that, if km, ne/o > Kne/o(Ξ½0), then ΟΞ½0e/o(m, n) β² eβkm, ne/o. Note that, although the inequality [87, Equation 3.7] in [87, Theorem 3.1] is stated for all m and n sufficiently large, the factor Dn on the right-hand side of [87, Equation 3.7] blows up if \(n\rightarrow \infty \), and thus exponential localisation is proved in [87] for fixed n as \(m\rightarrow \infty \).
Appendix F: Calculating the constant V loc in the Weyl asymptotics (3.9) for N loc
Recall from Section 3.4 that we need to compute the volume, Vloc, in phase space of the integrable tori contained entirely inside the small and large cavities and show that (3.10) holds.
Let
As in AppendixΒ E, let \(a:=\sqrt {{a_{1}^{2}}-{a_{2}^{2}}}\). We change variables π β [0, 2Ο] and \({\Omega } \in (0,\cosh ^{-1}(a_{1}/a))\)
(this is the same change of variables as (E.2) but with different variable names).
We now make a symplectic change of variables following [116, Β§2.3, Example 3 and Theorem 2.6]. If
then
and hence
Therefore, by [116, Theorem 2.6], the corresponding symplectomorphism is given by ΞΊ((Ο, π), (Οβ, πβ)) = ((x, y), (ΞΎ, Ξ·)) with
Then
so that, by [116, Theorem 2.10],
One can then easily check that
is conserved by the Hamiltonian flow of \(\widetilde {p}\); i.e. \(\dot {L}=0\).
Now, (F.2) and (F.3) imply that, on a billiard trajectory, \(\dot {\theta }=0\) if and only if πβ =β0, and \(\dot {\omega }=0\) if and only if Οβ =β0. That is, a trajectory is tangent to a curve {Ο = Ο0} when Οβ =β0, and to {π = π0} when πβ =β0. Note that
Next, observe that curves of constant π are confocal hyperbole, and curves of constant Ο are confocal ellipses. Since every trajectory is tangent to (possibly degenerate) confocal conic and L is constant, a trajectory is tangent to a (possibly degenerate) confocal ellipse if and only if L β₯β0 and to a confocal hyperbola if and only if L <β0. Since we are interested in the volume of phase space occupied by trajectories trapped near the minor axis, we consider the case L <β0. In that case, on \(\widetilde {p}=1\), the confocal hyperbola is given by
where we have used the fact that \(\sin \limits ^{2}(\theta )=-L\) and (F.1). We therefore want to find the volume of
in the (Ο, π, Οβ, πβ) variables, i.e. the volume of the set
We change variables in (Οβ, πβ) by letting
with Ο β [0, 2Ο], \(r\in [0,\infty )\). Then, \(\widetilde {p}=r\),
and
We now observe that
if and only if
if and only if
So
where Ξ¦ is defined by (3.11). Therefore, the volume of A equals
which is (3.10).
Finally, to determine the relevant Ξ±, we find the points where the boundary of the ellipse E (1.2) meets the hyperbola (F.4), i.e.
Rearranging the second inequality in (F.5), we find that
and so using the first inequality in (F.5), we find that
which implies that \(L= - y^{2} /{a_{2}^{2}}\). Therefore, if the ellipse is cut at ycut, then \(\alpha _{\text {cut}}= -y_{\text {cut}}^{2}/{a_{2}^{2}}\), i.e. (3.5) holds, and the volume of the relevant piece of phase space is indeed given by (3.10).
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Marchand, P., Galkowski, J., Spence, E.A. et al. Applying GMRES to the Helmholtz equation with strong trapping: how does the number of iterations depend on the frequency?. Adv Comput Math 48, 37 (2022). https://doi.org/10.1007/s10444-022-09931-9
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DOI: https://doi.org/10.1007/s10444-022-09931-9