An edge-centered scheme for anisotropic diffusion problems with discontinuities on distorted quadrilateral meshes

• Published in: Journal of computational science Vol. 64; p. 101832
• Main Authors: Wang, Yihong, Yang, Tinggan, Chang, Lina
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.10.2022
• Subjects: Anisotropic diffusion; Discontinuous diffusivity; Distorted quadrilateral meshes; Tailored Finite Point Method; Distorted quadrilateral meshes; Anisotropic diffusion; Discontinuous diffusivity; Tailored Finite Point Method
• ISSN: 1877-7503; 1877-7511
• DOI: 10.1016/j.jocs.2022.101832

We propose a Tailored Finite Point Method for the solution of transient anisotropic diffusion problems on distorted quadrilateral meshes. We establish an edge-centered scheme based on the flux continuity. The main feature of the scheme lies in the fact that the approximate solutions as well as the gradients are determined by using a linear combination of special solutions. These special solutions are required to satisfy the homogeneous form of approximated equations on the local cell. This brings two important assets to numerical solution of the problems. First, the scheme efficiently tailors itself fit into the local properties of the solution, so that it exhibits robustness to mesh distortion and can correctly reproduce the details of the interface layers even without refining the mesh. Second, the flux can be approximated in terms of the edge-centered unknowns defined at the local cell and the unknowns at the adjacent cells are not needed. The local stencil makes writing code for the continuity condition of the flux on each edge easier. Numerical experiments are carried out to validate the performance of the scheme with or without discontinuities on various types of distorted quadrilateral meshes. •We discuss the uniqueness of solutions, and its continuous dependence on sources and initial values of the transient anisotropic diffusion problem.•The main feature of the scheme lies in the fact that the approximate solution as well as the gradient are determined by using a linear combination of base functions.•An edge-centered scheme is obtained from the continuity of normal fluxes using exponential base functions.•Numerical results demonstrate that the new scheme is second order accurate for transient anisotropic diffusion problems on skewed meshes.•We do some numerical experiments to show the good accuracy and lack of sensitivity to mesh distortion with or without discontinuities based on exponential base functions.