An adaptive finite element DtN method for the elastic wave scattering problem in three dimensions

TL;DR

This paper introduces an adaptive finite element method with a Dirichlet-to-Neumann operator for 3D elastic wave scattering, efficiently handling boundary truncation and demonstrating exponential convergence and improved computational performance.

Contribution

It develops a novel adaptive finite element approach incorporating a DtN operator with exponential truncation error decay for 3D elastic scattering problems.

Findings

Exponential convergence of the truncation error with respect to the truncation parameter.

Efficient solution of the linear system using the generalized Woodbury matrix identity.

Numerical results show superior accuracy and efficiency of the proposed method.

Abstract

Consider the elastic scattering of an incident wave by a rigid obstacle in three dimensions, which is formulated as an exterior problem for the Navier equation. By constructing a Dirichlet-to-Neumann (DtN) operator and introducing a transparent boundary condition, the scattering problem is reduced equivalently to a boundary value problem in a bounded domain. The discrete problem with the truncated DtN operator is solved by using the a posteriori error estimate based adaptive finite element method. The estimate takes account of both the finite element approximation error and the truncation error of the DtN operator, where the latter is shown to converge exponentially with respect to the truncation parameter. Moreover, the generalized Woodbury matrix identity is utilized to solve the resulting linear system efficiently. Numerical experiments are presented to demonstrate the superior performance of the proposed method.