§06§ §16§Numerical simulation promises new insight in science and engineering. In ad dition to the traditional ways to perform research in science, that is laboratory experiments and theoretical work, a third way is being established: numerical simulation. It is based on both mathematical models and experiments con ducted on a computer. The discipline of scientific computing combines all aspects of numerical simulation. The typical approach in scientific computing includes modelling, numerics and simulation, see Figure l. Quite a lot of phenomena in science and engineering can be modelled by partial differential equations (PDEs). In order to produce accurate results, complex models and high resolution simulations are needed. While it is easy to increase the precision of a simulation, the computational cost of doing so is often prohibitive. Highly efficient simulation methods are needed to overcome this problem. This includes three building blocks for computational efficiency, discretisatio §16§n, solver and computer. Adaptive mesh refinement, high order and sparse grid methods lead to discretisations of partial differential equations with a low number of degrees of freedom. Multilevel iterative solvers decrease the amount of work per degree of freedom for the solution of discretised equation systems. Massively parallel computers increase the computational power available for a single simulation. §04§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. §02§Prof. Dr. Gerhard Zumbusch, Universität Jena
