Pressure-robust finite elements for the Stokes problem on three-dimensional curved domains

Abstract

This paper develops a divergence-free, inf-sup stable, optimally convergent and pressure-robust finite element method for the three-dimensional Stokes problem on curved domains. The geometry is approximated by an isoparametric tetrahedral mesh, while the velocity space is obtained from Scott--Vogelius spaces on an Alfeld split by the Piola transform. The discrete inf-sup condition is proved using suitable face bubble functions. The insufficient accuracy of quadrature rules on curved triangular interfaces leads to a consistency error that may cause suboptimal convergence. The remedy is to introduce a suitable consistency correction without stabilization terms. Moreover, commuting operators on curved domains are constructed from local commuting interpolants and shown to be globally conforming. The operators are used to approximate the load and obtain a pressure-robust discretization. Numerical examples are provided to validate the theoretical results.