Pressure-Robust Fortin-Soulie Elements of the Stokes Equation on Curved Domains

TL;DR

This paper develops a pressure-robust, divergence-free finite element method for the Stokes problem on curved domains, using isoparametric mapping and Piola transforms, with proven stability and optimal error estimates.

Contribution

It introduces a novel pressure-robust, divergence-free nonconforming finite element method for curved domains, enhancing stability and accuracy over existing approaches.

Findings

Proved inf-sup stability and optimal convergence rates.

Validated pressure-robustness through numerical examples.

Constructed elements via isoparametric mapping and Piola transform.

Abstract

This paper presents a pressure-robust and element-wise divergence-free nonconforming finite element method for the Stokes problem on curved domains. The discrete element is constructed by mapping the Fortin-Soulie element from a reference triangle using an isoparametric mapping for the geometry and a Piola transform for the function space. The inf-sup condition and the error estimate with optimal convergence rate are proved. Pressure-robustness is obtained by replacing the discrete velocity test functions with the first-order Raviart-Thomas functions. Numerical examples are provided to validate the theoretical results.