General degree divergence-free finite element methods for the Stokes problem on smooth domains

TL;DR

This paper develops divergence-free finite element methods for the Stokes problem on smooth domains, achieving optimal convergence rates in energy and $L^2$ norms through advanced finite element techniques.

Contribution

It introduces a new divergence-free finite element approach using Scott-Vogelius elements with arbitrary polynomial degree, ensuring optimal convergence on smooth domains.

Findings

Optimal order convergence in energy norm.

Optimal order convergence in $L^2$-norm.

Numerical experiments confirm theoretical results.

Abstract

In this paper, we construct and analyze divergence-free finite element methods for the Stokes problem on smooth domains. The discrete spaces are based on the Scott-Vogelius finite element pair of arbitrary polynomial degree greater than two. By combining the Piola transform with the classical isoparametric framework, and with a judicious choice of degrees of freedom, we prove that the method converges with optimal order in the energy norm. We also show that the discrete velocity error converges with optimal order in the $L^2$-norm. Numerical experiments are presented, which support the theoretical results.