A fully discrete Calderon Calculus for the two-dimensional elastic wave equation

TL;DR

This paper introduces a fully discrete boundary integral method for solving the 2D elastic wave equation, employing a non-conforming Petrov-Galerkin approach with precise quadrature and testing functions, validated through numerical tests.

Contribution

It presents a novel discretization scheme for elastic wave boundary integral operators, including a special treatment for the Hilbert transform in the kernel, with mathematical justification.

Findings

Accurate discretization of elastic layer potentials achieved.

Method successfully applied to frequency and time domain problems.

Demonstrated effectiveness on smooth open arcs.

Abstract

In this paper we present a full discretization of the layer potentials and boundary integral operators for the elastic wave equation on a parametrizable smooth closed curve in the plane. The method can be understood as a non-conforming Petrov-Galerkin discretization, with a very precise choice of testing functions by symmetrically combining elements on two staggered grids, and using a look-around quadrature formula. Unlike in the acoustic counterpart of this work, the kernel of the elastic double layer operator includes a periodic Hilbert transform that requires a particular choice of the mixing parameters. We give mathematical justification of this fact. Finally, we test the method on some frequency domain and time domain problems, and demonstrate its applicability on smooth open arcs.