Mimetic Finite Difference Methods for Diffusion Equations on Unstructured Triangular Grid

Summary

A finite difference algorithm for solution of stationary diffusion equation on unstructured triangular grid has been developed earlier by a support operator method. The support operator method first constructs a discrete divergence operator from the divergence theorem and then constructs a discrete gradient operator as the adjoint operator of the divergence. The adjointness of the operators is based on the continuum Gauss theorem which remains valid also for discrete operators. Here we extend the method to general Robin boundary conditions, generalize it to time dependent heat equation and perform the analysis of space discretization. One parameter family of discrete vector inner products, which produce exact gradients for linear functions, is designed. Our method works very well for discontinuous diffusion coefficient and very rough or very distorted grids which appear quite often e.g. in Lagrangian simulations.