We discuss a two-grid method for computing mixed finite-element approximations of nonlinear reaction-diffusion equations whose nonlinearity appears in the source term. This kind of equations arise in hydrology in the study of leaky aquifers and, perhaps more interestingly, in modeling nutrient diffusion in biofilms attached to porous media, and in the mathematical modeling of many biology phenomena, such as population dynamics.

We develop and analyze a two-grid technique for solving the nonlinear systems arising from implicit time discretization. In this technique, we decompose the large nonlinear system associated with a fine grid into two systems: a small nonlinear system on a coarse grid (mesh size H) and a larger linear system on a fine grid (mesh size h << H). Estimates show that the error is O(h^{k+1}+H^{2k+2}+\Delta t), where k is the degree of polynomials in the approximation spaces. These estimates guide the choice of coarse and fine mesh sizes to reduce the work invested in nonlinear iterations.

The matrix equations on both grids are saddle-point systems, which we solve using an efficient, multigrid-based scheme developed earlier. Both theoretical analysis and numerical experiments show the efficiency of the overall algorithm.