### A Two-Grid Method for Mixed Finite-Element

### Solution of Reaction-Diffusion Equations

#### Li Wu *
* Myron B. Allen

**Abstract**

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.