A Nonlinear Finite Volume Element Method Satisfying Maximum Principle for Anisotropic Diffusion Problems on Arbitrary Triangular Meshes (Communications in Computational Physics, 2019)
A nonlinear finite volume element scheme for anisotropic diffusion problems on general triangular meshes is proposed. Starting with a standard linear conforming finite volume element approximation, a corrective term with respect to theflux jumps across element boundaries is added to make the scheme satisfy the discrete maximum principle. The new scheme is free of the anisotropic non-obtuse anglecondition which is a severe restriction on the grids for problems with anisotropic diffusion. Moreover, this manipulation can nearly keep the same accuracy as the originalscheme. We prove the existence of the numerical solution for this nonlinear schemetheoretically. Numerical results and a grid convergence study are presented for bothcontinuous and discontinuous anisotropic diffusion problems.
Keywords: Finite volume element method; nonlinear correction; discrete maximum principle; anisotropic diffusion.