The elastic wave equation and the stable numerical coupling of its interior and exterior problems

Abstract

AbstractWe consider the elastic wave equation in three dimensions with transparent boundary conditions on a bounded, not necessarily convex domain. This leads to a coupling of the elastic wave equation in the interior domain with time‐dependent boundary integral equations. This coupled system is to be solved numerically. The major theoretical result of this paper shows a coercivity property of a Calderón operator for the elastic Helmholtz equation, which is valid for all complex frequencies in a half‐plane. Combining Laplace transform and energy techniques, this coercivity in the frequency domain is used to prove the stability of the fully discrete numerical method in the time domain. The considered numerical method couples finite elements and explicit leapfrog time‐stepping in the interior with boundary elements and convolution quadrature on the boundary. We present convergent error bounds for the semi‐ and full discretization.