Mixed finite element approximations based on 3‐Dhp‐adaptive curved meshes with two types ofH(div)‐conforming spaces

Abstract

SummaryTwo stable approximation space configurations are treated for the mixed finite element method for elliptic problems based on curved meshes. Their choices are guided by the property that, in the master element, the image of the flux space by the divergence operator coincides with the potential space. By using static condensation, the sizes of global condensed matrices, which are proportional to the dimension of border fluxes, are the same in both configurations. The meshes are composed of different topologies (tetrahedra, hexahedra, or prisms). Simulations using asymptotically affine uniform meshes, exactly fitting a spherical‐like region, and constant polynomial degree distributionk, showL2errors of orderk+1 ork+2 for the potential variable, while keeping orderk+1 for the flux in both configurations. The first case corresponds to RT(k) and BDFM(k+1) spaces for hexahedral and tetrahedral meshes, respectively, but holding for prismatic elements as well. The second case, further incrementing the order of approximation of the potential variable, holds for the three element topologies. The case ofhp‐adaptive meshes is considered for a problem modelling a porous media flow around a cylindrical horizontal well with elliptical drainage area. The effect of parallelism and static condensation in CPU time reduction is illustrated.