Nonlinear constrained finite element approximations to anisotropic diffusion problems
are considered. Starting with a standard (linear or bilinear) Galerkin discretization,
the entries of the stiffness matrix are adjusted so as to enforce sufficient
conditions of the discrete maximum principle (DMP). An algebraic splitting is employed
to separate the contributions of negative and positive off-diagonal coefficients
which are associated with diffusive and antidiffusive numerical fluxes, respectively.
In order to prevent the formation of spurious undershoots and overshoots, a symmetric
slope limiter is designed for the antidiffusive part. The corresponding upper
and lower bounds are 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 is performed by means of a lumped-mass L2 projection. The proposed
slope limiting strategy preserves 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 is devised for the nonlinear
algebraic system to be solved. Numerical results and a grid convergence study are
presented for a number of anisotropic diffusion problems in two space dimensions.