Inf-sup stability of the trace P2-P1 Taylor-Hood elements for surface PDEs

TL;DR

This paper proves the inf-sup stability of the trace P2-P1 Taylor-Hood finite element pair for surface PDEs, ensuring reliable numerical solutions on complex geometries with uniform stability bounds.

Contribution

It establishes the first inf-sup stability proof for trace P2-P1 Taylor-Hood elements on surface PDEs, with stability constants independent of surface position.

Findings

Uniform inf-sup stability proven for trace P2-P1 elements.

Optimal order convergence demonstrated for the method.

Numerical examples confirm theoretical properties.

Abstract

The paper studies a geometrically unfitted finite element method (FEM), known as trace FEM or cut FEM, for the numerical solution of the Stokes system posed on a closed smooth surface. A trace FEM based on standard Taylor-Hood (continuous P2-P1) bulk elements is proposed. A so-called volume normal derivative stabilization, known from the literature on trace FEM, is an essential ingredient of this method. The key result proved in the paper is an inf-sup stability of the trace P2-P1 finite element pair, with the stability constant uniformly bounded with respect to the discretization parameter and the position of the surface in the bulk mesh. Optimal order convergence of a consistent variant of the finite element method follows from this new stability result and interpolation properties of the trace FEM. Properties of the method are illustrated with numerical examples.