High Order Cut Finite Elements for the Elastic Wave Equation

TL;DR

This paper develops a high order cut finite element method for the elastic wave equation that remains stable and accurate even when boundaries cut through the mesh arbitrarily, using stabilization and Nitsche's method.

Contribution

It introduces a novel high order cut finite element approach with stabilization and symmetric formulations for elastic wave problems involving complex boundaries.

Findings

Method achieves optimal order accuracy in 2D.

Stability is maintained regardless of boundary cuts.

Numerical experiments confirm theoretical error estimates.

Abstract

A high order cut finite element method is formulated for solving the elastic wave equation. Both a single domain problem and an interface problem are treated. The boundary or interface are allowed to cut through the background mesh. To avoid problems with small cuts, stabilizing terms are added to the bilinear forms corresponding to the mass and stiffness matrix. The stabilizing terms penalize jumps in normal derivatives over the faces of the elements cut by the boundary/interface. This ensures a stable discretization independently of how the boundary/interface cuts the mesh. Nitsche's method is used to enforce boundary and interface conditions, resulting in symmetric bilinear forms. As a result of the symmetry, an energy estimate can be made and optimal order a priori error estimates are derived for the single domain problem. Finally, numerical experiments in two dimensions are presented that verify the order of accuracy and stability with respect to small cuts.