Theory and Applications of Multigrid and Domain Decomposition Methods

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There are three areas of research in the project ``Theory and Applications of Multigrid and Domain Decomposition Methods''. The first area concerns the additive multigrid convergence theory, which is effective in analyzing the asymptotic behavior of the contraction numbers of multigrid algorithms with respect to the number of smoothing steps for boundary value problems with less than full elliptic regularity. The seminal result obtained by the PI for V-cycle algorithm with Richardson relaxation as the smoother will be extended to general smoothers, nonconforming finite elements, the mixed formulation and the F-cycle algorithm. The second area involves multigrid methods for stress intensity factors and singular solutions. These are methods that can take advantage of the form of the solution of the boundary value problem in the regions where it does not have full elliptic regularity. Using this approach, usual quasi-optimal convergence rates have been obtained for simple finite elements on simple grids for boundary value problems on two-dimensional domains with reentrant corners or cracks. The technically more challenging interface problems and three dimensional problems will be investigated in this project. The third area is the analysis of the Finite Element Tearing and Interconnecting (FETI) method, a nonoverlapping domain decomposition method in which the (pseudo) inverse of the Schur complement matrix has to be preconditioned. The goal of this part of the project is to carry out an analysis of the FETI method and some of its variants within the framework of additive Schwarz preconditioners, and to investigate new mechanisms for the global communication among the subdomains. The methods analyzed in this project are efficient algorithms for the numerical solution of partial differential equations. Such equations are extremely important in science and engineering since they are the governing equations for most physical phenomena. Part of the research involves the fast computation of stress intensity factors, which are essential indicators in fracture mechanics. The FETI method to be studied in this project has already been implemented for large scale engineering problems using parallel supercomputers with up to a thousand processors. The advances resulting from this project will therefore have an impact on many areas of science and technology, such as aerospace engineering, fracture prediction and fluid flow problems.