Divergence--free Scott--Vogelius elements on curved domains

TL;DR

This paper introduces a divergence-free Scott-Vogelius finite element method on curved domains for the Stokes problem, ensuring optimal convergence, pressure robustness, and applicability to curved geometries.

Contribution

It develops an isoparametric finite element pair using a Piola transform that maintains divergence-free properties on curved domains, extending Scott-Vogelius elements.

Findings

Converges with optimal order

Is divergence-free and pressure robust

Supported by numerical examples

Abstract

We construct and analyze an isoparametric finite element pair for the Stokes problem in two dimensions. The pair is defined by mapping the Scott-Vogelius finite element space via a Piola transform. The velocity space has the same degrees of freedom as the quadratic Lagrange finite element space, and therefore the proposed spaces reduce to the Scott-Vogelius pair in the interior of the domain. We prove that the resulting method converges with optimal order, is divergence--free, and is pressure robust. Numerical examples are provided which support the theoretical results.