Convex combinations for diffusion schemes

An interpolation method for diffusion in anisotropic discontinuous media based on convex combinations and physical relationships is presented. The method is exact for any piecewise linear solution, even if the interpolation nodes lie on the opposite sides of a material discontinuity. Values in points that do not lie within the convex hull of interpolation nodes are computed using flux boundary conditions. The method permits interpolation in every point within most domains, while preserving the maximum and minimum principles. We propose to replace the interpolation techniques in several non-linear finite volume schemes with the present method. Additionally, it is demonstrated that the construction of a convex combination by a simple search strategy may not be economical or even feasible if the grid is distorted. An alternative search structure that behaves well in such cases is proposed.