Eigenvalues of the truncated Helmholtz solution operator under strong trapping

For the Helmholtz equation posed in the exterior of a Dirichlet obstacle, we prove that if there exists a family of quasimodes (as is the case when the exterior of the obstacle has stable trapped rays), then there exist near-zero eigenvalues of the standard variational formulation of the exterior Dirichlet problem (recall that this formulation involves truncating the exterior domain and applying the exterior Dirichlet-to-Neumann map on the truncation boundary). The significance of this result is a) the finite-element method for computing approximations to solutions of the Helmholtz equation is based on the standard variational formulation, and b) the location of eigenvalues, and especially near-zero ones, plays a key role in understanding how iterative solvers such as the generalised minimum residual method (GMRES) behave when used to solve linear systems, in particular those arising from the finite-element method. The result proved in this paper is thus the first step towards rigorously understanding how GMRES behaves when applied to discretisations of high-frequency Helmholtz problems under strong trapping (the subject of the companion paper [MGSS21]).

[1]  Vladimir F. Lazutkin,et al.  Kam Theory and Semiclassical Approximations to Eigenfunctions , 1993 .

[2]  Jeffrey Galkowski,et al.  High-frequency estimates on boundary integral operators for the Helmholtz exterior Neumann problem , 2021, ArXiv.

[3]  Vicente Hernández,et al.  SLEPc: A scalable and flexible toolkit for the solution of eigenvalue problems , 2005, TOMS.

[4]  S. Dyatlov,et al.  Dynamical zeta functions for Anosov flows via microlocal analysis , 2013, 1306.4203.

[5]  William Gropp,et al.  Efficient Management of Parallelism in Object-Oriented Numerical Software Libraries , 1997, SciTools.

[6]  Yogi A. Erlangga,et al.  Advances in Iterative Methods and Preconditioners for the Helmholtz Equation , 2008 .

[7]  F. Brezzi,et al.  On the coupling of boundary integral and finite element methods , 1979 .

[8]  Stéphanie Chaillat,et al.  Fast iterative boundary element methods for high-frequency scattering problems in 3D elastodynamics , 2017, J. Comput. Phys..

[9]  Lars Hr̲mander,et al.  The Analysis of Linear Partial Differential Operators III: Pseudo-Differential Operators , 1985 .

[10]  Martin J. Gander,et al.  A Class of Iterative Solvers for the Helmholtz Equation: Factorizations, Sweeping Preconditioners, Source Transfer, Single Layer Potentials, Polarized Traces, and Optimized Schwarz Methods , 2016, SIAM Rev..

[11]  P. Stefanov,et al.  Resonances near the real axis imply existence of quasimodes , 2000 .

[12]  Maciej Zworski,et al.  From quasimodes to resonances , 1998 .

[13]  András Vasy.  Propagation of singularities for the wave equation on manifolds with corners , 2005 .

[14]  Jeffrey Galkowski,et al.  Local absorbing boundary conditions on fixed domains give order-one errors for high-frequency waves , 2021, ArXiv.

[15]  Cornelis Vuik,et al.  On a Class of Preconditioners for Solving the Helmholtz Equation , 2003 .

[16]  A. Jennings.  Influence of the Eigenvalue Spectrum on the Convergence Rate of the Conjugate Gradient Method , 1977 .

[17]  P. Stefanov,et al.  Neumann resonances in linear elasticity for an arbitrary body , 1996 .

[18]  Barry F. Smith,et al.  PETSc Users Manual , 2019 .

[19]  Luc Miller,et al.  Refraction of high-frequency waves density by sharp interfaces and semiclassical measures at the boundary , 2000 .

[20]  Francisco-Javier Sayas,et al.  Relaxing the hypotheses of Bielak–MacCamy’s BEM–FEM coupling , 2012, Numerische Mathematik.

[21]  Stephen Langdon,et al.  Numerical-asymptotic boundary integral methods in high-frequency acoustic scattering* , 2012, Acta Numerica.

[22]  Plamen Stefanov,et al.  Distribution of resonances for the Neumann problem in linear elasticity outside a strictly convex body , 1995 .

[23]  Maciej Zworski,et al.  Resonance expansions of scattered waves , 2000 .

[24]  S. Dyatlov,et al.  Fractal Weyl laws and wave decay for general trapping , 2017, 1703.06515.

[25]  Dianne P. O'Leary,et al.  Eigenanalysis of some preconditioned Helmholtz problems , 1999, Numerische Mathematik.

[26]  Ilse C. F. Ipsen,et al.  GMRES and the minimal polynomial , 1996 .

[27]  Martin J. Gander,et al.  How Large a Shift is Needed in the Shifted Helmholtz Preconditioner for its Effective Inversion by Multigrid? , 2017, SIAM J. Sci. Comput..

[28]  Martin J. Coen,et al.  With O'Leary? , 1984 .

[29]  Preprint Mps.  Condition Number Estimates for Combined Potential Integral Operators in Acoustics and their Boundary Element Discretisation , 2010 .

[30]  Frédéric Hecht,et al.  New development in freefem++ , 2012, J. Num. Math..

[31]  Dianne P. O'Leary,et al.  A Multigrid Method Enhanced by Krylov Subspace Iteration for Discrete Helmholtz Equations , 2001, SIAM J. Sci. Comput..

[32]  David Lafontaine,et al.  A sharp relative-error bound for the Helmholtz h-FEM at high frequency , 2019, Numerische Mathematik.

[33]  Semiclassical analysis , 2019, Graduate Studies in Mathematics.

[34]  Andrew J. Wathen,et al.  Performance and analysis of saddle point preconditioners for the discrete steady-state Navier-Stokes equations , 2002, Numerische Mathematik.

[35]  Ke Chen,et al.  Efficient preconditioners for iterative solution of the boundary element equations for the three-dimensional Helmholtz equation , 2001 .

[36]  Christophe Geuzaine,et al.  Double sweep preconditioner for optimized Schwarz methods applied to the Helmholtz problem , 2014, J. Comput. Phys..

[37]  Jens Markus Melenk,et al.  Convergence analysis for finite element discretizations of the Helmholtz equation with Dirichlet-to-Neumann boundary conditions , 2010, Math. Comput..

[38]  James Demmel,et al.  SuperLU_DIST: A scalable distributed-memory sparse direct solver for unsymmetric linear systems , 2003, TOMS.

[39]  Y. Saad,et al.  GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems , 1986 .

[40]  Maarten V. de Hoop,et al.  Solving the Three-Dimensional High-frequency Helmholtz Equation Using Contour Integration and Polynomial Preconditioning , 2020, SIAM J. Matrix Anal. Appl..

[41]  Cornelis Vuik,et al.  Spectral Analysis of the Discrete Helmholtz Operator Preconditioned with a Shifted Laplacian , 2007, SIAM J. Sci. Comput..

[42]  Jeffrey Galkowski,et al.  Optimal constants in nontrapping resolvent estimates and applications in numerical analysis , 2018, Pure and Applied Analysis.

[43]  Martin J. Gander,et al.  Why it is Difficult to Solve Helmholtz Problems with Classical Iterative Methods , 2012 .

[44]  E. A. Spence,et al.  Domain Decomposition with Local Impedance Conditions for the Helmholtz Equation with Absorption , 2018, SIAM J. Numer. Anal..

[45]  Alastair Spence,et al.  Applying GMRES to the Helmholtz equation with strong trapping: how does the number of iterations depend on the frequency? , 2021, Advances in Computational Mathematics.

[46]  S. Dyatlov,et al.  Mathematical Theory of Scattering Resonances , 2019, Graduate Studies in Mathematics.

[47]  Marion Darbas,et al.  Combining analytic preconditioner and Fast Multipole Method for the 3-D Helmholtz equation , 2013, J. Comput. Phys..

[48]  Plamen Stefanov,et al.  Quasimodes and resonances: Sharp lower bounds , 1999 .