Adaptive finite volume schemes for anisotropic heterogeneous diffusion problems on arbitary convex and nonconvex meshes

Solving the anisotropic heterogeneous diffusion problems requires a well adopted subdivision of the computational domain with a mesh refinement procedure to numerically guarantee the convergence. A new refinement procedure for quadrilateral meshes in convex or non-convex cases is elaborated herein. Various DDFV schemes are elaborated for anisotropic, strongly anisotropic and discontinuous diffusion tensors. Structured and unstructured meshes are considered in the convex and non-convex elements as well as arbitrary meshes with various distortion orders. The presented procedure allows eliminating the convergence rate sensitivity. The main advantage of this mesh refinement strategy is to reduce the percentage of non-convex elements at each step after refinement and to achieve super-convergence on discrete H1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H<^>{1}$$\end{document} and L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L<^>{2}$$\end{document}-norms. The efficiency and performance of the elaborated adaptive DDFV scheme are numerically demonstrated for the considered quadrilateral meshes.