Approximation of H(div) with High-Order Optimal Finite Elements for Pyramids, Prisms and Hexahedra

Classical facet elements do not provide an optimal rate of convergence of the numerical solution toward the solution of the exact problem in H(div)-norm for general unstructured meshes containing hexahedra and prisms. We propose two new families of high-order elements for hexahedra, triangular prisms and pyramids that recover the optimal convergence. These elements have compatible restrictions with each other, such that they can be used directly on general hybrid meshes. Moreover the H(div) proposed spaces are completing the De Rham diagram with optimal elements previously constructed for H and H(curl) approximation. The obtained pyramidal elements are compared theoretically and numerically with other elements of the literature. Eventually, numerical results demonstrate the efficiency of the finite elements constructed.