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 Th∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {T}_h^*$$\end{document} and its submesh Th∗∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {T}_h^{**}$$\end{document} from a primal mesh Th\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {T}_h$$\end{document} in such a manner that a set number of adjacent tetrahedral elements of Th∗∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {T}_h^{**}$$\end{document} are united to form each dual control volume of Th∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {T}_h^*$$\end{document}, which corresponds to a primal vertex. The weak solution of the diffusion problem is approximated by the piecewise linear functions on the subdual mesh Th∗∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {T}_h^{**}$$\end{document}. 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.