A divergence-free finite element method for the Stokes problem with boundary correction

TL;DR

This paper introduces a boundary correction finite element method for the Stokes problem that ensures divergence-free velocity approximation and optimal convergence, using Scott-Vogelius elements with boundary modifications.

Contribution

It develops a divergence-free finite element method with boundary correction for the Stokes problem, ensuring well-posedness and optimal convergence.

Findings

The method achieves divergence-free velocity approximation.

The method converges with optimal order.

The boundary correction improves boundary condition enforcement.

Abstract

This paper constructs and analyzes a boundary correction finite element method for the Stokes problem based on the Scott-Vogelius pair on Clough-Tocher splits. The velocity space consists of continuous piecewise quadratic polynomials, and the pressure space consists of piecewise linear polynomials without continuity constraints. A Lagrange multiplier space that consists of continuous piecewise quadratic polynomials with respect to boundary partition is introduced to enforce boundary conditions as well as to mitigate the lack of pressure-robustness. We prove several inf-sup conditions, leading to the well-posedness of the method. In addition, we show that the method converges with optimal order and the velocity approximation is divergence free.