H(div) finite elements based on nonaffine meshes for 3D mixed formulations of flow problems with arbitrary high order accuracy of the divergence of the flux

Abstract

Effects of nonaffine elements on the accuracy of 3D H(div)-conforming finite elements are studied. Instead of convergence order k+1 for the flux and the divergence of the flux obtained with Raviart-Thomas or Nedelec spaces with normal traces of degree k, based on affine hexahedra or triangular prisms, reduced orders k for the flux and k-1 for the divergence of the flux may occur for distorted elements. To improve this scenario, a hierarchy of enriched flux approximations is considered, by adding internal shape functions up to a higher degree k+n, n>0, while keeping the original normal traces of degree k. The resulting enriched approximations, using multilinear transformations, keep the original flux accuracy (of order k+1 with affine elements or reduced order k otherwise), but enhanced divergence (of order k+n+1, in the affine case, or k+n-1 otherwise) can be reached. The reduced flux accuracy due to quadrilateral face distortions cannot be corrected by including higher order internal functions. The enriched spaces are applied to the mixed finite element formulation of Darcy's model. The computational cost of matrix assembly increases with n, but the condensed system to be solved has the same dimension and structure as the original scheme