Two-Grid Discretization Scheme for Nonlinear Reaction Diffusion Equation by Mixed Finite Element Methods (Advances in Applied Mathematics and Mechanics, 2014)
In this paper, we study an efficient scheme for nonlinear reaction-diffusionequations discretized by mixed finite element methods. We mainly concern the casewhen pressure coefficients and source terms are nonlinear. To linearize the nonlinearmixed equations, we use the two-grid algorithm. We first solve the nonlinear equationson the coarse grid, then, on the fine mesh, we solve a linearized problem usingNewton iteration once. It is shown that the algorithm can achieve asymptotically optimalapproximation as long as the mesh sizes satisfy $H =\mathcal{O}(h^{\frac{1}{2}})$. As a result, solvingsuch a large class of nonlinear equations will not be much more difficult than gettingsolutions of one linearized system.