On surface meshes induced by level set functions

The zero level set of a piecewise-affine function with respect to a consistent tetrahedral subdivision of a domain in R3 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. We show that the nodal basis of a P1 finite element space with respect to this surface triangulation is L2-stable, provided a natural scaling is used. Furthermore, the issue of stability of the nodal basis with respect to the H1-norm is addressed.