Applying GMRES to the Helmholtz equation with strong trapping: how does the number of iterations depend on the frequency?

• Published in: Advances in computational mathematics Vol. 48; no. 4
• Main Authors: Marchand, P., Galkowski, J., Spence, E. A., Spence, A.
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.08.2022; Springer Nature B.V; Springer Verlag
• Subjects: Advances in Computational Integral Equations; Computational mathematics; Computational Mathematics and Numerical Analysis; Computational Science and Engineering; Dirichlet problem; Eigenvalues; Helmholtz equations; Integral equations; Mathematical and Computational Biology; Mathematical Modeling and Industrial Mathematics; Mathematics; Mathematics and Statistics; Numerical Analysis; Trapping; Upper bounds; Visualization; Helmholtz equation; Trapping; High frequency; GMRES; trapping; high-frequency
• ISSN: 1019-7168; 1572-9044
• DOI: 10.1007/s10444-022-09931-9

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.