Multilevel Fictitious Space Preconditioner for Nonconforming Approximations on Unstructured Regular Triangulations

In the multidimensional numerical simulation of physical processes many phenomena are suuciently localized and it is obvious that adap-tive local grid reenement techniques are necessary to resolve the local physical behavior. For this reason the nite element discretiza-tions are often considered on the non-hierarchical unstructured meshes which permit to reduce the number of degrees of freedom in comparison with regular structured mesh maintaining the quality of approximation. For large-scale simulation problems involving unstructured meshes and possible local grid reenement, an eeciency is the key to the choice of solution technique.Among various methods for solving partial diierential equations, multigrid or multilevel methods have proven to be one of the most eecient approaches. However, the ee-ciency of these methods depends crucially on appropriate underlying multilevel structure. Since such multilevel structures, or hierarchy, are not naturally available in most unstructured grids, multigrid methods are not easy to apply in these cases. In recent years many non-nested multigrid methods have been developed for unstructured grids 1, 2, 3]. A common idea in these non-nested methods is to make use of a non-nested sequence of grids by same sophisticated coarsening techniques. Another approach was proposed in 6, 7]. That technique is based on the ctitious space method, i.e. the reduction of the original problem to a problem in an auxiliary ((ctitious) space, and the multilevel