Parallel Multilevel Methods: Adaptive Mesh Refinement and Loadbalancing

ISBN-13: 9783519004516 / Angielski / Miękka / 2003 / 216 str.

Gerhard Zumbusch

Opis

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, discretisation, 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."

Spis treści

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.

O autorze

Prof. Dr. Gerhard Zumbusch, Universität Jena

Blurb

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.