An adaptive finite element DtN method for the three-dimensional acoustic scattering problem

TL;DR

This paper develops an adaptive finite element method with a truncated Dirichlet-to-Neumann operator for 3D acoustic scattering, providing error estimates and demonstrating efficiency through numerical experiments.

Contribution

It introduces an a posteriori error estimate for the truncated DtN-based finite element method and develops an adaptive algorithm for 3D acoustic scattering problems.

Findings

Error estimate shows exponential decay of truncation error

Adaptive method effectively refines mesh based on error estimates

Numerical results confirm the method's efficiency and accuracy

Abstract

This paper is concerned with a numerical solution of the acoustic scattering by a bounded impenetrable obstacle in three dimensions. The obstacle scattering problem is formulated as a boundary value problem in a bounded domain by using a Dirichlet-to-Neumann (DtN) operator. An a posteriori error estimate is derived for the finite element method with the truncated DtN operator. The a posteriori error estimate consists of the finite element approximation error and the truncation error of the DtN operator, where the latter is shown to decay exponentially with respect to the truncation parameter. Based on the a posteriori error estimate, an adaptive finite element method is developed for the obstacle scattering problem. The truncation parameter is determined by the truncation error of the DtN operator and the mesh elements for local refinement are marked through the finite element approximation error. Numerical experiments are presented to demonstrate the effectiveness of the proposed method.