Implicit Extrapolation Methods for Variable Coefficient Problems

SUMMARY Implicit extrapolation methods for the solution of partial diierential equations are based on applying the extrapolation principle indirectly. Multigrid tau-extrapolation is a special case of this idea. In the context of multilevel nite element methods, an algorithm of this type can be used to raise the approximation order, even when the meshes are nonuniform or locally reened. Here previous results are generalized to the variable coeecient case and thus become applicable for nonlinear problems. The implicit extrapolation multigrid algorithm converges to the solution of a higher order nite element system. This is obtained without explicitly constructing higher order stiiness matrices but by applying extrapolation in a natural form within the algorithm. The algorithm requires only a small change of a basic low order multigrid method.