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

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