Discrete strong extremum principles for finite element solutions of diffusion problems with nonlinear corrections

Abstract

A nonlinear correction technique for finite element methods to anisotropic diffusion problems on general triangular and quadrilateral meshes are introduced. The classic linear or bi-linear finite element methods are modified, and then the resulting schemes satisfy the discrete strong extremum principle unconditionally, which means that it is unnecessary to impose the well-known restrictions on diffusion coefficients and geometry of mesh-cell (e.g. “acute angle” condition). Convergence rate for smooth and piecewise smooth solutions and the discrete extremum principle property are verified by numerical examples.