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arXiv · 1301.3745

On surface meshes induced by level set functions

Maxim A. Olshanskii·Arnold Reusken·Xianmin Xu
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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.

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Maxim A. Olshanskii, Arnold Reusken, Xianmin Xu. 2013-01-16. On surface meshes induced by level set functions. https://arxiv.org/abs/1301.3745

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