Analysis of a high order Trace Finite Element Method for PDEs on level set surfaces

TL;DR

This paper introduces a high order trace finite element method for PDEs on level set surfaces, providing error analysis, stabilization techniques, and numerical validation for improved accuracy and stability.

Contribution

It presents a novel high order trace finite element approach with an isoparametric mapping, along with a unified stabilization analysis and numerical validation.

Findings

Achieves optimal order $H^1( abla)$-norm error bounds.

Identifies stabilization method based on anisotropic diffusion as most effective.

Numerical results confirm theoretical error bounds and condition number control.

Abstract

We present a new high order finite element method for the discretization of partial differential equations on stationary smooth surfaces which are implicitly described as the zero level of a level set function. The discretization is based on a trace finite element technique. The higher discretization accuracy is obtained by using an isoparametric mapping of the volume mesh, based on the level set function, as introduced in [C. Lehrenfeld, \emph{High order unfitted finite element methods on level set domains using isoparametric mappings}, Comp. Meth. Appl. Mech. Engrg. 2016]. The resulting trace finite element method is easy to implement. We present an error analysis of this method and derive optimal order $H^1(\Gamma)$-norm error bounds. A second topic of this paper is a unified analysis of several stabilization methods for trace finite element methods. Only a stabilization method which is based on adding an anisotropic diffusion in the volume mesh is able to control the condition number of the stiffness matrix also for the case of higher order discretizations. Results of numerical experiments are included which confirm the theoretical findings on optimal order discretization errors and uniformly bounded condition numbers.