Construction of 4D Simplex Space-Time Meshes for Local Bisection Schemes

TL;DR

This paper introduces a method for constructing 4D simplex space-time meshes with relaxed preconditions, enabling efficient local bisection refinement in higher-dimensional finite element methods.

Contribution

It proves that 4-colorable tetrahedral meshes can be used to generate 4D simplex meshes satisfying bisection preconditions, extending mesh refinement capabilities.

Findings

4D simplex meshes can be constructed from 4-colorable tetrahedral meshes.

The method relaxes strict preconditions for simplex bisection in higher dimensions.

Strategies are discussed for handling non-4-colorable tetrahedral meshes.

Abstract

Adaptive mesh refinement is a key component of efficient unstructured space-time finite element methods. Underlying any adaptive mesh refinement scheme is, of course, a method for local refinement of simplices. However, simplex bisection algorithms in dimension greater than three have strict mesh preconditions which can be hard to satisfy. We prove that certain four-dimensional simplex space-time meshes can be handled with a relaxed precondition. Namely, we prove that if a tetrahedral mesh is 4-colorable, then we can produce a 4D simplex mesh which always satisfies the bisection precondition. We also briefly discuss strategies to handle tetrahedral meshes which are not 4-colorable.