Main aspects of the efficient treatment of partial differential equations are discretisation, multilevel/multigrid solution and parallelisation. These distinct topics are coverd from the historical background to modern developments. It is demonstrated how the ingredients can be put together to give an adaptive and parallel multilevel approach for the solution of elliptic boundary value problems. Error estimators and adaptive grid refinement techniques for ordinary and for sparse grid discretisations are presented. Different types of additive and multiplicative multilevel solvers are discussed with respect to parallel implementation and application to adaptive refined grids. Efficiency issues are treated both for the sequential multilevel methods and for the parallel version by hash table storage techniques. Finally, space-filling curve enumeration for parallel load balancing and processor cache efficiency are discussed.

Prof. Dr. Gerhard Zumbusch, Universität Jena

1 Introduction.- 2 Multilevel Iterative Solvers.- 2.1 Direct and Iterative Solvers.- 2.2 Subspace Correction Schemes.- 2.3 Multigrid and Multilevel Methods.- 2.4 Domain Decomposition Methods.- 2.5 Sparse Grid Solvers.- 3 Adaptively Refined Meshes.- 3.1 The Galerkin Method, Finite Elements and Finite Differences.- 3.2 Error Estimation and Adaptive Mesh Refinement.- 3.3 Data Structures for Adaptively Refined Meshes.- 4 Space-Filling Curves.- 4.1 Definition and Construction.- 4.2 Partitioning.- 4.3 Partitions of Adaptively Refined Meshes.- 4.4 Partitions of Sparse Grids.- 5 Adaptive Parallel Multilevel Methods.- 5.1 Multigrid on Adaptively Refined Meshes.- 5.2 Parallel Multilevel Methods.- 5.3 Parallel Adaptive Methods.- 6 Numerical Applications.- 6.1 Parallel Multigrid for a Poisson Problem.- 6.2 Parallel Multigrid for Linear Elasticity.- 6.3 Parallel Solvers for Sparse Grid Discretisations.- Concluding Remarks and Outlook.