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There are different possibilities of choosing balanced pairs of approximation spaces for dual (flux) and primal (pressure) variables; to be used in discrete versions of the mixed finite element method for elliptic problems arising in fluid simulations. Three cases shall be studied for discretized three dimensional formulations, based on tetrahedral, hexahedral, and prismatic meshes. The principle guiding the constructions of the approximation spaces is the property that, the divergence of the dual space and the primal approximation space, should coincide, while keeping the same order of accuracy for the flux variable, and varying the accuracy order of the primal variable. Some cases correspond either to the classic spaces of Raviart-Thomas, Brezzi-Douglas-Marini, Brezzi-Douglas-Fortin-Marini or Nedelec types. A new kind of approximation is proposed by further incrementing the order of some internal flux functions, and matching primal functions at the border fluxes. In this article we develop a unified error analysis for all these space families, and element geometries.

Error analysis for a new mixed finite element method in 3D