In this paper we present two hierarchically preconditioned methods for the fast solution of mesh equations that approximate 2D-elliptic boundary value problems on unstructured quasi uniform triangulations. Based on the fictitious space approach the original problem can be embedded into an auxiliary one, where both the hierarchical grid information and the preconditioner by decomposing functions on it are well defined. We implemented the corresponding Yserentant preconditioned conjugate gradient method as well as the BPX-preconditioned cg-iteration having optimal computational costs. Several numerical examples demonstrate the efficiency of the artificially constructed hierarchical methods which can be of importance in the industrial engineering, where often only the nodal coordinates and the element connectivity of the underlying (fine) discretization are available.

In this paper we present two hierarchically preconditioned methods for the fast solution of mesh equations that approximate 2D-elliptic boundary value problems on unstructured quasi uniform triangulations. Based on the fictitious space approach the original problem can be embedded into an auxiliary one, where both the hierarchical grid information and the preconditioner by decomposing functions on it are well defined. We implemented the corresponding Yserentant preconditioned conjugate gradient method as well as the BPX-preconditioned cg-iteration having optimal computational costs. Several numerical examples demonstrate the efficiency of the artificially constructed hierarchical methods which can be of importance in the industrial engineering, where often only the nodal coordinates and the element connectivity of the underlying (fine) discretization are available.

Extrapolation methods for the solution of partial differential equations 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 \034 -extrapolation. For finite element systems this algorithm can be shown to be equivalent to higher order finite 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 efficient multigrid and multilevel preconditioning techniques. The effectivity of the method is demonstrated for heat conduction problems and problems from elasticity theory.

Extrapolation methods for the solution of partial differential equations 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 \034 -extrapolation. For finite element systems this algorithm can be shown to be equivalent to higher order finite 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 efficient multigrid and multilevel preconditioning techniques. The effectivity of the method is demonstrated for heat conduction problems and problems from elasticity theory.