Non-linear constrained finite element approximations to anisotropic diffusion problems were described. Starting with a standard (linear or bilinear) Galerkin discretization, the entries of the stiffness matrix were adjusted to enforce sufficient conditions on the discrete maximum principle. Algebraic splitting was employed to separate the contributions of negative and positive off-diagonal coefficients which were associated with diffusive and anti-diffusive numerical fluxes, respectively. To prevent spurious under- and over-shoots, a symmetrical slope limiter was designed for the anti-diffusive part. The corresponding upper and lower bounds were defined using an estimate of the steepest gradient in terms of the maximum and minimum solution values at surrounding nodes. The recovery of nodal gradients was performed using a lumped-mass L2 projection. The proposed slope limiting strategy preserved the consistency of the underlying discrete problem and the structure of the stiffness matrix (symmetry, zero row and column sums). A positivity-preserving defect correction scheme was devised for the non-linear algebraic system to be solved. Numerical results and a grid convergence study were presented for a number of anisotropic diffusion problems in 2 dimensions.

A Constrained Finite Element Method Satisfying the Discrete Maximum Principle for Anisotropic Diffusion Problems. D.Kuzmin, M.J.Shashkov, D.Svyatskiy: Journal of Computational Physics, 2009, 228[9], 3448-63