A Divergence-free finite element method for the stokes problem with boundary correction

Abstract

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 polynomials of degree k, and the pressure space consists of piecewise polynomials of degree (k − 1) without continuity constraints. A Lagrange multiplier space that consists of continuous piecewise polynomials with respect to the boundary partition is introduced to enforce boundary conditions and 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.