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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.

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
Full Text: DOI
[1] Becker, Acta Numerica (2001)
[2] Braack, Stabilized finite elements for 3-D reactive flows, International Journal for Numerical Methods in Fluids 51 pp 981– (2006) · Zbl 1158.80326 · doi:10.1002/fld.1160
[3] Prohl, Projection and Quasi-Compressibility Methods for Solving the Incompressible Navier-Stokes Equations (1997) · doi:10.1007/978-3-663-11171-9
[4] Becker R Braack M Meidner D Richter T Vexler B http://www.gascoigne.uni-hd.de
[5] Ciarlet, Finite Element Methods for Elliptic Problems (1978)
[6] Braack, Solutions of 3D Navier-Stokes benchmark problems with adaptive finite elements, Computers and Fluids 35 (4) pp 372– (2006) · Zbl 1160.76364 · doi:10.1016/j.compfluid.2005.02.001
[7] Becker, Multigrid techniques for finite elements on locally refined meshes, Numerical Linear Algebra with Applications 7 pp 373– (2000) · Zbl 1051.65117 · doi:10.1002/1099-1506(200009)7:6<363::AID-NLA202>3.0.CO;2-V
[8] Brandt, Multigrid Solvers on Parallel Computers (1981) · doi:10.1016/B978-0-12-632620-8.50008-5
[9] Xu, Iterative methods by space decomposition and subspace correction, SIAM Review 34 (4) pp 581– (1992) · Zbl 0788.65037 · doi:10.1137/1034116
[10] Richter T Parallel multigrid method for adaptive finite elements with application to 3D flow problems 2005 · Zbl 1082.76002
[11] Bastian, Parallele Adaptive Mehrgitterverfahren (1996) · doi:10.1007/978-3-322-99572-8
[12] Schupp B Entwicklung eines effizienten Verfahrens zur Simulation kompressibler Strömungen in 3D auf Parallelrechnern 1999
[13] Braack, International Conference on Hyperbolic Problems: Theory, Numerics, Applications 140 pp 169– (2001) · doi:10.1007/978-3-0348-8370-2_18
[14] Olbers, Chorin Workshop on Stochastic Climate Models pp 3– (2001) · doi:10.1007/978-3-0348-8287-3_1
[15] Griffies, Ocean modelling with MOM, CLIVAR Exchanges 12 (3) pp 3– (2007)
[16] Severijns, The efficient global primitive equation climate model SPEEDO, Geoscientific Model Development 2 pp 1115– (2009) · doi:10.5194/gmdd-2-1115-2009
[17] Pedlosky, Geophysical Fluid Dynamics (1986)
[18] Berselli, Mathematics of Large Scale Simulation of Turbulent Flows (2006) · Zbl 1089.76002
[19] Guermond, An overview of projection methods for incompressible flows, Computer Methods in Applied Mechanics and Engineering 195 pp 6011– (2006) · Zbl 1122.76072 · doi:10.1016/j.cma.2005.10.010
[20] Rannacher, The Navier-Stokes Equations II-Theory and Numerical Methods pp 167– (1992) · doi:10.1007/BFb0090341
[21] Helics 2007 http://helics.iwr.uni-heidelberg.de
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