Parallel Multilevel Methods Based on Domain Decomposition Ideas

Bodo Heise Institute for Mathematics, Johannes Kepler University Linz Altenberger Str. 69, A-4040 Linz, Austria

                e-mail: heise@miraculix.numa.uni-linz.ac.at

Phone: +43-732-2468-9157; Fax: +43-732-2468-10

Michael Jung Faculty for Mathematics, Technical University Chemnitz-Zwickau D-09107 Chemnitz, Germany

               e-mail: michael.jung@mathematik.tu-chemnitz.de

Phone: +49-371-531-2163; Fax: +49-371-531-2657

In our talk, the solution of two-dimensional nonlinear elliptic boundary value problems on parallel machines with Multiple Instruction Multiple Data (MIMD) architecture is discussed. Especially, we consider electro–magnetic field problems the numerical solution of which is based on finite element discretizations and a nested Newton solver.

For solving the linear systems of algebraic finite element equations in each Newton step, parallel conjugate gradient methods with a Domain Decomposition preconditioner (DD PCG) as well as parallelized global multilevel methods are applied. The latter include the global BPX preconditioner for a CG method (BPX PCG) as well as a parallelized global multigrid method (GMG). Further, the GMG method can be used as a preconditioner in the CG algorithm which accelerates convergence, especially if multigrid rates are poor, and makes multigrid more robust.

The non–overlapping Domain Decomposition data structure which has been created initially for local DD PCG methods allows us to implement the global methods mentioned above with low communication effort, too. Therefore, the implementation of the whole algorithm, i.e. the mesh generation, the generation of the finite element equations, the nested Newton algorithm, and the solvers, is based on the non–overlapping DD data structure.

In our numerical results, we compare the efficiency of the different solvers for model problems and for technical problems of practical relevance as, e.g., electric machines.