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

Morgane Bergot (1) , Marc Duruflé (2, 3)

1 CALVI - Scientific computation and visualization
2 IMB - Institut de Mathématiques de Bordeaux
3 BACCHUS - Parallel tools for Numerical Algorithms and Resolution of essentially Hyperbolic problems

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 H1 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.

Mots clés: Facet Elements; High-order finite element; Pyramids; H(div) approximation; De Rham diagram

Domaines: Equations aux dérivées partielles [math.AP]

Morgane Bergot, Marc Duruflé. Approximation of H(div) with High-Order Optimal Finite Elements for Pyramids, Prisms and Hexahedra. Communications in Computational Physics, 2013, 14 (5), pp.1372-1414. ⟨10.4208/cicp.120712.080313a⟩. ⟨hal-00723472v2⟩