On surface meshes induced by level set functions

TL;DR

This paper proves that zero level sets of piecewise-affine functions on tetrahedral meshes can be postprocessed into well-conditioned surface triangulations, enabling accurate PDE solutions on these surfaces.

Contribution

It introduces a method to convert level set zero surfaces into high-quality triangulations with optimal error bounds for PDE discretization.

Findings

Surface triangulation satisfies maximum angle condition after postprocessing.

Diagonal scaling yields well-conditioned mass matrices uniformly in mesh size.

Optimal interpolation error bounds are derived for PDE solutions on these surfaces.

Abstract

The zero level set of a piecewise-affine function with respect to a consistent tetrahedral subdivision of a domain in $\mathbb{R}^3$ is a piecewise-planar hyper-surface. We prove that if a family of consistent tetrahedral subdivions satisfies the minimum angle condition, then after a simple postprocessing this zero level set becomes a consistent surface triangulation which satisfies the maximum angle condition. We treat an application of this result to the numerical solution of PDEs posed on surfaces, using a $P_1$ finite element space on such a surface triangulation. For this finite element space we derive optimal interpolation error bounds. We prove that the diagonally scaled mass matrix is well-conditioned, uniformly with respect to $h$. Furthermore, the issue of conditioning of the stiffness matrix is addressed.