Quantitative bounds on Impedance-to-Impedance operators with applications to fast direct solvers for PDEs

We prove quantitative norm bounds for a family of operators involving impedance boundary conditions on convex, polygonal domains. A robust numerical construction of Helmholtz scattering solutions in variable media via the Dirichlet-to-Neumann operator involves a decomposition of the domain into a sequence of rectangles of varying scales and constructing impedance-to-impedance boundary operators on each subdomain. Our estimates in particular ensure the invertibility, with quantitative bounds in the frequency, of the merge operators required to reconstruct the original Dirichlet-to-Neumann operator in terms of these impedance-to-impedance operators of the sub-domains. A key step in our proof is to obtain Neumann and Dirichlet boundary trace estimates on solutions of the impedance problem, which are of independent interest. In addition to the variable media setting, we also construct bounds for similar merge operators in the obstacle scattering problem.

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

[2]  Timo Betcke,et al.  An Exponentially Convergent Nonpolynomial Finite Element Method for Time-Harmonic Scattering from Polygons , 2010, SIAM J. Sci. Comput..

[3]  Peter Monk,et al.  An analysis of the coupling of finite-element and Nyström methods in acoustic scattering , 1994 .

[4]  Q. Fang,et al.  Viable Three-Dimensional Medical Microwave Tomography: Theory and Numerical Experiments , 2010, IEEE Transactions on Antennas and Propagation.

[5]  Daniel Tataru,et al.  ON THE REGULARITY OF BOUNDARY TRACES FOR THE WAVE EQUATION , 1998 .

[6]  C. Turc,et al.  Schur complement Domain Decomposition Methods for the solution of multiple scattering problems , 2016, 1608.00034.

[7]  Jared Wunsch,et al.  Sharp High-Frequency Estimates for the Helmholtz Equation and Applications to Boundary Integral Equations , 2015, SIAM J. Math. Anal..

[8]  A. Majda,et al.  Absorbing boundary conditions for the numerical simulation of waves , 1977 .

[9]  A. Kirsch,et al.  Convergence analysis of a coupled finite element and spectral method in acoustic scattering , 1990 .

[10]  Peter Monk,et al.  A least-squares method for the Helmholtz equation , 1999 .

[11]  Eddie Wadbro,et al.  High contrast microwave tomography using topology optimization techniques , 2010, J. Comput. Appl. Math..

[12]  M. Zworski,et al.  Geometric control in the presence of a black box , 2004 .

[13]  Andrew Hassell,et al.  Fast Computation of High‐Frequency Dirichlet Eigenmodes via Spectral Flow of the Interior Neumann‐to‐Dirichlet Map , 2011, 1112.5665.

[14]  Mitsuru Ikawa.  On the poles of the scattering matrix for two strictly convex obstacles , 1983 .

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

[16]  Rusell Brown,et al.  The mixed problem for laplace's equation in a class of lipschitz domains , 1994 .

[17]  A. Nachman,et al.  Global uniqueness for a two-dimensional inverse boundary value problem , 1996 .

[18]  Sonia Fliss,et al.  Exact boundary conditions for periodic waveguides containing a local perturbation , 2006 .

[19]  C. Gérard.  Asymptotique des pôles de la matrice de scattering pour deux obstacles strictement convexes , 1986 .

[20]  Jens Markus Melenk,et al.  FEM-BEM mortar coupling for the Helmholtz problem in three dimensions , 2020, Comput. Math. Appl..

[21]  Per-Gunnar Martinsson,et al.  A Direct Solver with O(N) Complexity for Variable Coefficient Elliptic PDEs Discretized via a High-Order Composite Spectral Collocation Method , 2013, SIAM J. Sci. Comput..

[22]  N. Burq.  Contrôle de l'équation de Schrödinger en présence d'obstacles strictement convexes , 1991 .

[23]  P. Grisvard.  Elliptic Problems in Nonsmooth Domains , 1985 .

[24]  Hans Christianson,et al.  Neumann Data Mass on Perturbed Triangles. , 2019, 1908.02863.

[25]  O. R. Pembery,et al.  The Helmholtz equation in heterogeneous media: A priori bounds, well-posedness, and resonances , 2018, Journal of Differential Equations.

[26]  Jianlin Xia,et al.  On 3D modeling of seismic wave propagation via a structured parallel multifrontal direct Helmholtz solver , 2011 .

[27]  J. Descloux,et al.  An accurate algorithm for computing the eigenvalues of a polygonal membrane , 1983 .

[28]  Hans Christianson,et al.  Equidistribution of Neumann data mass on triangles , 2017, 1701.02793.

[29]  Per-Gunnar Martinsson,et al.  A spectrally accurate direct solution technique for frequency-domain scattering problems with variable media , 2013, 1308.5998.

[30]  A. Majda,et al.  Radiation boundary conditions for acoustic and elastic wave calculations , 1979 .

[31]  Nicholas Hale,et al.  The ultraspherical spectral element method , 2020, ArXiv.

[32]  Bruno Després,et al.  A Domain Decomposition Method for the Helmholtz equation and related Optimal Control Problems , 1996 .

[33]  A. Barnett,et al.  Comparable upper and lower bounds for boundary values of Neumann eigenfunctions and tight inclusion of eigenvalues , 2015, Duke Mathematical Journal.

[34]  C. Bardos,et al.  Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary , 1992 .

[35]  Dirk Klindworth,et al.  Robin-to-Robin transparent boundary conditions for the computation of guided modes in photonic crystal wave-guides , 2015 .

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

[37]  M. Gunzburger,et al.  Boundary conditions for the numerical solution of elliptic equations in exterior regions , 1982 .

[38]  Jeffrey Galkowski,et al.  Sharp norm estimates of layer potentials and operators at high frequency , 2014, 1403.6576.

[39]  Per-Gunnar Martinsson,et al.  Fast Direct Solvers for Elliptic PDEs , 2019 .