Efficient Solution of Heterogeneous Anisotropic Diffusion Problems

Diffusion equation with anisotropic coefficients arise in many environmental topics, heat transfer, groundwater flow and transport problems, petroleum reservoir simulations, hydrodynamic simulations, ... These problems are characterized by a full rank diffusion tensor, that is diagonal only if the reference system is aligned with the principal direction of anisotropy (Bear). Numerical difficulties in solving this type of equation could arise when anisotropy, the tendency of a phenomenon to progressively align along a preferential direction, becomes strong. A new methodology for the solution of the anisotropic heterogeneous diffusion problem is presented in this paper. The governing equations are discretized over a basic unstructured triangular mesh satisfying the so called Generalized Delaunay condition, further specified. The resulting spatial discretization of the fluxes across the control volume is similar to the one occurring in the standard linear (P1) Galerkin Finite Element scheme.