TY - JOUR

T1 - DECOMPOSITIONS OF HIGH-FREQUENCY HELMHOLTZ SOLUTIONS VIA FUNCTIONAL CALCULUS, AND APPLICATION TO THE FINITE ELEMENT METHOD

AU - Galkowski, J.

AU - Lafontaine, D.

AU - Spence, E. A.

AU - Wunsch, J.

N1 - Publisher Copyright:
© 2023 Society for Industrial and Applied Mathematics Publications. All rights reserved.

PY - 2023

Y1 - 2023

N2 - Over the last 10 years, results from [J. M. Melenk and S. Sauter, Math. Comp., 79 (2010), pp. 1871-1914], [J. M. Melenk and S. Sauter, SIAM J. Numer. Anal., 49 (2011), pp. 1210- 1243], [S. Esterhazy and J. M. Melenk, Numerical Analysis of Multiscale Problems, Springer, New York, 2012, pp. 285-324] and [J. M. Melenk, A. Parsania, and S. Sauter, J. Sci. Comput., 57 (2013), pp. 536-581] decomposing high-frequency Helmholtz solutions into "low-" and "high-" frequency components have had a large impact in the numerical analysis of the Helmholtz equation. These results have been proved for the constant-coefficient Helmholtz equation in either the exterior of a Dirichlet obstacle or an interior domain with an impedance boundary condition. Using the Helffer- Sjöstrand functional calculus [B. Helffer and J. Sjöstrand, Schrödinger Operators, Springer, Berlin, 1989, pp. 118-197] this paper proves analogous decompositions for scattering problems fitting into the black-box scattering framework of Sjöstrand and Zworski [J. Amer. Math. Soc., 4 (1991), pp. 729-769] thus covering Helmholtz problems with variable coefficients, impenetrable obstacles, and penetrable obstacles all at once. These results allow us to prove new frequency-explicit convergence results for (i) the hp-finite-element method (hp-FEM) applied to the variable-coefficient Helmholtz equation in the exterior of an analytic Dirichlet obstacle, where the coefficients are analytic in a neighborhood of the obstacle, and (ii) the h-FEM applied to the Helmholtz penetrable-obstacle transmission problem. In particular, the result in (i) shows that the hp-FEM applied to this problem does not suffer from the pollution effect.

AB - Over the last 10 years, results from [J. M. Melenk and S. Sauter, Math. Comp., 79 (2010), pp. 1871-1914], [J. M. Melenk and S. Sauter, SIAM J. Numer. Anal., 49 (2011), pp. 1210- 1243], [S. Esterhazy and J. M. Melenk, Numerical Analysis of Multiscale Problems, Springer, New York, 2012, pp. 285-324] and [J. M. Melenk, A. Parsania, and S. Sauter, J. Sci. Comput., 57 (2013), pp. 536-581] decomposing high-frequency Helmholtz solutions into "low-" and "high-" frequency components have had a large impact in the numerical analysis of the Helmholtz equation. These results have been proved for the constant-coefficient Helmholtz equation in either the exterior of a Dirichlet obstacle or an interior domain with an impedance boundary condition. Using the Helffer- Sjöstrand functional calculus [B. Helffer and J. Sjöstrand, Schrödinger Operators, Springer, Berlin, 1989, pp. 118-197] this paper proves analogous decompositions for scattering problems fitting into the black-box scattering framework of Sjöstrand and Zworski [J. Amer. Math. Soc., 4 (1991), pp. 729-769] thus covering Helmholtz problems with variable coefficients, impenetrable obstacles, and penetrable obstacles all at once. These results allow us to prove new frequency-explicit convergence results for (i) the hp-finite-element method (hp-FEM) applied to the variable-coefficient Helmholtz equation in the exterior of an analytic Dirichlet obstacle, where the coefficients are analytic in a neighborhood of the obstacle, and (ii) the h-FEM applied to the Helmholtz penetrable-obstacle transmission problem. In particular, the result in (i) shows that the hp-FEM applied to this problem does not suffer from the pollution effect.

KW - FEM

KW - Helmholtz

KW - hp-FEM

KW - splitting

UR - http://www.scopus.com/inward/record.url?scp=85174357825&partnerID=8YFLogxK

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U2 - 10.1137/21M1409160

DO - 10.1137/21M1409160

M3 - Article

AN - SCOPUS:85174357825

SN - 0036-1410

VL - 55

SP - 3903

EP - 3958

JO - SIAM Journal on Mathematical Analysis

JF - SIAM Journal on Mathematical Analysis

IS - 4

ER -