An enhanced flux continuity three-dimensional finite element method for heterogeneous and anisotropic diffusion problems on general meshes

In this research, we present a novel enhanced flux continuity three-dimensional finite element method for heterogeneous and anisotropic (possibly discontinuous) diffusion problems on general meshes. We create a polygonal dual mesh \(\mathcal {T}_h^*\) ${\mathcal{T}}_h^{\ast }$ and its submesh \(\mathcal {T}_h^{**}\) ${\mathcal{T}}_h^{\ast \ast }$ from a primal mesh \(\mathcal {T}_h\) ${\mathcal{T}}_h$ in such a manner that a set number of adjacent tetrahedral elements of \(\mathcal {T}_h^{**}\) ${\mathcal{T}}_h^{\ast \ast }$ are united to form each dual control volume of \(\mathcal {T}_h^*\) ${\mathcal{T}}_h^{\ast }$ , which corresponds to a primal vertex. The weak solution of the diffusion problem is approximated by the piecewise linear functions on the subdual mesh \(\mathcal {T}_h^{**}\) ${\mathcal{T}}_h^{\ast \ast }$ . In order to capture the local continuity of numerical fluxes across the interfaces, the proposed scheme gives the auxiliary face unknowns interpolated by the multi-point flux approximation. Moreover, the consistency, coercive, and convergence properties of the method are presented within a rigorous theoretical framework. Numerical results are carried out to highlight accuracy and efficiency.