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Parallel multigrid method for finite element simulations of complex flow problems on locally refined meshes. (English) Zbl 1265.76043
The article describes a parallel solution algorithm for realistic complex three-dimensional flow problems. It concentrates on the ‘numerical linear algebra’ techniques to solve the linearized equations obtained by finite element discretizations on locally refined meshes generated by an adaptive algorithm. Such an algorithm is undoubtedly necessary to obtain good computation efficiency and has recently become an important ingredient of modern software in the field. The authors clearly work out the challenges for the linear solution algorithms arising from local mesh refinement: strongly heterogenous meshes, which in addition vary during the overall solution procedure. This requires adequate data structures and algorithms with optimal complexity, which in the context of parallel computing is a highly non-trivial task. It should be noted that the literate on fast parallel solvers nearly exclusively treats the case of quasi-uniform (or even structured) meshes. The article under review therefore strongly contributes to close this gap.
The reader finds a detailed description of the ingredients of the proposed parallel algorithm. Some theoretical hints for the behavior of the parallel smoother, as compared to the sequential variant, are also given. Finally, two challenging examples (simulation of a gas burner and ocean flow) are presented. They clearly show the convincing scaling of the proposed algorithm.

MSC:
76M10 Finite element methods applied to problems in fluid mechanics
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs
65Y05 Parallel numerical computation
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