Implicit Extrapolation Methods for Multilevel Finite Element Computations

Extrapolation methods for the solution of partial diierentialequations are commonly based on the existence of error expansions for the approximate solution. Implicit extrapolation, in the contrast, is based on applying extrapolation indirectly, by using it on quantities like the residual. In the context of multigrid methods, a special technique of this type is known as-extrapolation. For nite element systems this algorithm can be shown to be equivalent to higher order nite elements. The analysis is local and does not use global expansions, so that the implicit extrapolation technique may be used on unstructured meshes and in cases where the solution fails to be globally smooth. Furthermore, the natural multilevel structure can be used to construct eecient multigrid and multilevel preconditioning techniques. The eeectivity of the method is demonstrated for heat conduction problems and problems from elasticity theory.