Boundary elements with mesh refinements for the wave equation

TL;DR

This paper develops boundary element methods with mesh refinements for solving the wave equation in polyhedral domains, addressing singularities at corners and edges to improve numerical accuracy.

Contribution

It formulates boundary integral equations for the wave equation in time domain and demonstrates the effectiveness of graded meshes in achieving optimal approximation rates.

Findings

Graded meshes recover optimal approximation rates for singular solutions.

Numerical experiments validate the theoretical results.

Applications include Dirichlet, Neumann problems, and sound emission modeling.

Abstract

The solution of the wave equation in a polyhedral domain in $\mathbb{R}^3$ admits an asymptotic singular expansion in a neighborhood of the corners and edges. In this article we formulate boundary and screen problems for the wave equation as equivalent boundary integral equations in time domain, study the regularity properties of their solutions and the numerical approximation. Guided by the theory for elliptic equations, graded meshes are shown to recover the optimal approximation rates known for smooth solutions. Numerical experiments illustrate the theory for screen problems. In particular, we discuss the Dirichlet and Neumann problems, as well as the Dirichlet-to-Neumann operator and applications to the sound emission of tires.