A priori error estimates for compatible spectral discretization of the Stokes problem for all admissible boundary conditions

Abstract

This paper describes the recently developed mixed mimetic spectral element method for the Stokes problem in the vorticity-velocity-pressure formulation. This compatible discretization method relies on the construction of a conforming discrete Hodge decomposition, that is based on a bounded projection operator that commutes with the exterior derivative. The projection operator is the composition of a reduction and a reconstruction step. The reconstruction in terms of mimetic spectral element basis-functions are tensor-based constructions and therefore hold for curvilinear quadrilateral and hexahedral meshes. For compatible discretization methods that contain a conforming discrete Hodge decomposition, we derive optimal a priori error estimates which are valid for all admissible boundary conditions on both Cartesian and curvilinear meshes. These theoretical results are confirmed by numerical experiments. These clearly show that the mimetic spectral elements outperform the commonly used H(div)-compatible Raviart-Thomas elements.