Tailored Finite Point Method for Diffusion Equations with Interfaces on Distorted Meshes

Diffusion processes is usually coupled with other physical processes such as the fluid equation. The meshes are determined by the fluid that can be distorted as time goes on. Classical finite difference schemes and finite element method are sensitive of mesh deformation. We propose a new tailored finite point method (TFPM) for 2D diffusion equation with tensor diffusion coefficient on highly distorted meshes. Second order convergence is demonstrated numerically with and without interfaces. TFPM is a finite difference method that makes full use of the analytical properties of local solutions. The main advantages of TFPM is that no modifications have to be made for problems with strongly discontinuous coefficients, where most other methods require special treatment at the interfaces. This advantage is important for distorted meshes, since the designing of numerical discretizations near interfaces is more delicate for distorted meshes.