Linearly scalable fast direct solver based on proxy surface method for two-dimensional elastic wave scattering by cavity

Abstract

This paper proposes an $O\left(N\right)$ fast direct solver for two-dimensional elastic wave scattering problems. The proxy surface method is extended to elastodynamics to obtain shared coefficients for low-rank approximations from discretized integral operators. The proposed method is a variant of the Martinsson–Rokhlin–type fast direct solver. Our variant avoids the explicit computation of the inverse of the coefficient matrix, thereby reducing the required number of matrix–matrix multiplications. Numerical experiments demonstrate that the proposed solver has a complexity of $O\left(N\right)$ in the low-frequency range and has a highly parallel computation efficiency with a strong scaling efficiency of 70%. Furthermore, multiple right-hand sides can be solved efficiently; specifically, when solving problems with 180 right-hand side vectors, the processing time per vector from the second vector onward was approximately 28,900 times faster than that for the first vector. This is a key advantage of fast direct methods.