Primal finite element solution of second order problems in three‐dimension space with normal stress/flux continuity

Abstract

A tetrahedral finite element method of the Hermite type for solving second order elliptic equations in three‐dimension bounded domains is introduced. It is a sort of extension of the Morley triangle [3] but contrary to this element it provides converging approximations in this case. The new element is particularly useful in situations where flux continuity across interelement boundaries is required.

Hermite finite elements for second order boundary value problems with sharp gradient discontinuities

“…Using arguments in all similar to those in [3,4] we can assert that problem (4) has a unique solution. Moreover, according to [3,4] the broken H 1 -seminorm ∥ · ∥ h0 given by…”

“…In [3,4] it was established that B is an invertible matrix for o i = n i , that is, for K = αI where α ∈ ℜ and α > 0 (isotropic case). It follows that if K varies continuously and is sufficiently close to a tensor of that form, then by continuity and standard arguments B will also be invertible.…”

“…We exhibited in [4,3] According to [4,3], the canonical basis functions for the isotropic case are suitable combinations of a set of N(N + 1)/2 quadratic functions associated with the faces or edges of T ∈ T h . Since this will also be the case in the anisotropic case, we recall these functions below.…”

“…This method denoted by HP 2 was defined in [3,4] for the isotropic case, i.e. for K = αI, α ∈ ℜ, α > 0.…”

“…The first method studied here was introduced in [3] for three-dimensional problems and in [4] for two-dimensional ones. It is based on a Hermite representation of the primal unknown field with complete quadratics defined in each element of a triangular or a tetrahedral mesh.…”