A Stable Cut Finite Element Method for Partial Differential Equations on Surfaces: The Helmholtz-Beltrami Operator

TL;DR

This paper introduces a stable finite element method for solving the Helmholtz equation on surfaces embedded in 3D, ensuring stability and optimal error estimates even when the mesh does not conform to the surface.

Contribution

The paper develops a stabilized cut finite element method combining Galerkin least squares and gradient jump penalties for surface Helmholtz problems, with proven stability and error bounds.

Findings

Stability under mesh size and wave number condition $h k < C$

Optimal error estimates in $H^1$ and $L^2$ norms

Method effective on non-conforming tetrahedral meshes

Abstract

We consider solving the surface Helmholtz equation on a smooth two dimensional surface embedded into a three dimensional space meshed with tetrahedra. The mesh does not respect the surface and thus the surface cuts through the elements. We consider a Galerkin method based on using the restrictions of continuous piecewise linears defined on the tetrahedra to the surface as trial and test functions.Using a stabilized method combining Galerkin least squares stabilization and a penalty on the gradient jumps we obtain stability of the discrete formulation under the condition $h k < C$, where $h$ denotes the mesh size, $k$ the wave number and $C$ a constant depending mainly on the surface curvature $\kappa$, but not on the surface/mesh intersection. Optimal error estimates in the $H^1$ and $L^2$-norms follow.