Scott-Vogelius element and iterated penalty method for inhomogeneous Dirichlet boundary conditions

Abstract

We present quasi-optimal a priori error estimates for general mixed finite element methods to approximate solutions of the Stokes problem subject to inhomogeneous Dirichlet boundary conditions. For the Scott-Vogelius element this yields pressure-robust a priori error estimates. Due to the exact divergence constraint, this requires a compatibility condition for the boundary data to hold. A key tool is a modified Fortin operator, capable of preserving this compatibility condition. Furthermore, we analyse the iterated penalty method, a Uzawa-type algorithm and we show its convergence and asymptotic pressure robustness. Numerical experiments support the theory and highlight the importance of the compatibility condition and the appropriate treatment of nearly singular vertices.