# TraceFEM for the membrane problem using distance functions on $P_{1}$ and $P_{2}$ tetrahedra

## TL;DR

This paper develops and evaluates Trace finite element methods for the membrane problem on second order tetrahedral elements, focusing on level set reconstruction, stabilization, and numerical convergence analysis.

## Contribution

It introduces a stabilization technique for higher order membrane models and compares different element orders and level set functions in a comprehensive numerical study.

## Key findings

- Stabilization improves solution accuracy for higher order models
- Level set functions based on geometrical distance reduce errors
- Numerical convergence results validate the proposed methods

## Abstract

We consider Trace finite element methods for the linear membrane problem on second order tetrahedral elements. To accomplish this, zero-level set reconstruction methods for second order tetrahedra are considered. For the higher order membrane model a corresponding stabilization is proposed and numerically evaluated. We compare combinations of background- and surface element order and provide numerical convergence results. The impact of the stabilization on the resulting solution is numerically analyzed. We also compare the choice of level set function with respect to the geometrical distance and normal errors.