Two-Grid Methods for Mixed Finite Element Approximations of Nonlinear Parabolic Equations

Mixed finite element approximation of nonlinear parabolic equations is discussed. The equation considered is a prototype of a model which arises in flow through porous media. A two-grid approximation scheme is developed and analyzed for implicit time discretizations. In this approach, the full nonlinear system is solved on a "coarse" grid of size H. The nonlinearities are expanded about the coarse grid solution, and the resulting linear but nonsymmetric system is solved on a "fine" grid of size h. Error estimates are derived which demonstrate that the error is O (h^{k+1} + H^{2(k+1)-d/2} + Delta­t), where k is the degree of the approximating space for the primary variable and d is spatial dimension, with k >= 1 for d >= 2. For the RT0 space (k=0) on rectangular domains, we present a modified scheme for treating the coarse grid problem. Here we show that the error is O (h + H^{3 - d/2} + Delta­t). The above estimates are useful for determining an appropriate H for the coarse grid problem