Generating admissible space-time meshes for moving domains in $d+1$-dimensions

TL;DR

This paper introduces a novel method for generating admissible space-time meshes for moving domains in higher dimensions, enabling accurate finite element solutions of transient Stokes equations.

Contribution

It presents an algorithm to generate $d+1$-dimensional simplex space-time meshes for moving geometries, ensuring mesh admissibility and facilitating numerical simulations.

Findings

Successfully generates 4D meshes for moving domains

Demonstrates the method on transient Stokes equations

Provides initial numerical results validating the approach

Abstract

In this paper we present a discontinuous Galerkin finite element method for the solution of the transient Stokes equations on moving domains. For the discretization we use an interior penalty Galerkin approach in space, and an upwind technique in time. The method is based on a decomposition of the space-time cylinder into finite elements. Our focus lies on three-dimensional moving geometries, thus we need to triangulate four dimensional objects. For this we will present an algorithm to generate $d+1$-dimensional simplex space-time meshes and we show under natural assumptions that the resulting space-time meshes are admissible. Further we will show how one can generate a four-dimensional object resolving the domain movement. First numerical results for the transient Stokes equations on triangulations generated with the newly developed meshing algorithm are presented.