Abstract
In this paper we discuss different possibilities of using partially ordered sets of grids in multigrid algorithms. Because, for a classical sequence of regular grids the number of degrees of freedom grows much faster with the refinement level for 3D than for 2D, it is more difficult to find sufficiently effective relaxation procedures. Therefore, we study the possibility of using different families of (regular rectangular) grids.
Semi-coarsening is one technique in which a partially ordered set of grids is used. In this case still a unique discrete fine-grid problem is solved. On the other hand, sparse grid techniques are more efficient if we compare the accuracy obtained with the number of degrees of freedom used. However, in the latter case it is not always straightforward to identify an appropriate discrete equation that should be solved. The different approaches are compared.
The relation between the different approaches is described by looking at hierarchical bases and by considering full approximation (FAS). We show that, by lack of a semi-orthogonality property, the 3D situation is essentially more difficult than the 2D case. We also describe different multigrid strategies. Numerical results are given for a transonic Euler-flow over the ONERA M6-wing.
AMS Subject Classification (1991): 65N50, 65N55, 65N99
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References
A. Brandt. Guide to multigrid development. In W. Hackbusch and U. Trottenberg, editors, Multigrid Methods, volume 960 of Lecture Notes in Mathematics, pages 220–312. Springer-Verlag, Berlin, 1982.
H. J. Bungartz. Dünne Gitter und deren Anwendung bei der adaptiven Lösung der dreidimensionalen Poisson-Gleichung. PhD thesis, Institut für Informatik, TU München, 1992.
M. Griebel, M. Schneider, and C. Zenger. A combination technique for the solution of sparse grid problems. In Proceedings of the IMACS International Symposium on Iterative Methods in Linear Algebra, pages 263–281, Amsterdam, 1992. Elsevier.
M. Griebel, C. Zenger, and S. Zimmer. Multilevel Gauss-Seidel-algorithms for full and sparse grid problems. Computing, 50:127–148, 1993.
P. W. Hemker. Sparse-grid finite-volume multigrid for 3D-problems. Adv. Comput. Math., 4:83–110, 1995.
P. W. Hemker and B. Koren. A non-linear multigrid method for the steady Euler equations. In A. Dervieux, B. van Leer, J. Périaux, and A. Rizzi, editors, Numerical Simulation of Compressible Euler Flows, volume 26 of Notes on Numerical Fluid Mechanics, pages 175–196. Vieweg & Son, Braunschweig, Germany, 1989.
P. W. Hemker and S. P. Spekreijse. Multiple grid and Osher’s scheme for the efficient solution of the steady Euler equations. Appl. Numer. Math., 2:458–476, 1986.
P.W. Hemker and P.M. de Zeeuw. BASIS3, a data structure for 3-dimensional sparse grids. In H. Deconinck and B. Koren, editors, Euler and Navier-Stokes Solvers Using Multi-Dimensional Upwind Schemes and Multigrid Acceleration, volume 56 of Notes on Numerical Fluid Mechanics, pages 443–484. Vieweg, Braunschweig, 1997.
P. W. Hemker and C. Pflaum. Approximation on partially ordered sets of regular grids. Applied Numerical Mathematics, 25:55–87, 1997.
B. Koren. Defect correction and multigrid for an efficient and accurate computation of airfoil flows. J. Comput. Phys., 77:183–206, 1988.
B. Koren, P.W. Hemker, and C.T.H. Everaars. Multiple semi-coarsened multigrid for 3D CFD. In: Proceedings of the 13 th AIAA Computational Fluid Dynamics Conference, Snowmass Village, CO, pages 892–902 (AIAA-paper 97-2029), American Institute of Aeronautics and Astronautics, Reston, VA, 1997.
B. Koren, P.W. Hemker, and P.M. de Zeeuw. Semi-coarsening in three directions for Euler-flow computations in three dimensions. In: H. Deconinck and B. Koren, editors, Euler and Navier-Stokes Solvers Using Multi-Dimensional Upwind Schemes and Multigrid Acceleration, volume 56 of Notes on Numerical Fluid Mechanics, pages 547–567. Vieweg, Braunschweig, 1997.
C.B. Liem, T. Lu, and T.M. Shih. The Splitting Extrapolation Method. World Scientific, Singapore, 1995.
W. A. Mulder. A new multigrid approach to convection problems. J. Comput. Phys., 83:303–323, 1989.
N. H. Naik and J. R. van Rosendale. The improved robustness of multigrid elliptic solvers based on multiple semicoarsened grids. SIAM J. Numer. Anal., 30:215–229, 1993.
C. Pflaum. A multi-level-algorithm for the finite-element-solution of general second order elliptic differential equations on adaptive sparse grids. Technical Report TUM-I-9419, SFB-Bericht Nr. 342/12/94 A, Technische Universität München. Institut für Informatik, May 1994.
R. Radespiel and R. C. Swanson. Progress with multigrid schemes for hypersonic flow problems. J. Comput. Phys., 116, 1995.
U. Rüde. Multilevel, extrapolation, and sparse grid methods. In Multigrid Methods IV, Proceedings of the Fourth European Multigrid Conference, Amsterdam, July 6–9, 1993, P.W. Hemker and P. Wesseling eds. Volume 116 of ISNM, pages 281–294, Basel, 1994. Birkhäuser.
C. Zenger. Sparse grids. In Parallel algorithms for partial differential equations: Proceedings of the Sixth GAMM-Seminar, Kiel, Jan. 1990, Notes on Numerical Fluid Mechanics, vol. 31. Vieweg, Braunschweig, 1991.
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Hemker, P.W., Koren, B., Noordmans, J. (1998). 3D Multigrid on Partially Ordered Sets of Grids. In: Hackbusch, W., Wittum, G. (eds) Multigrid Methods V. Lecture Notes in Computational Science and Engineering, vol 3. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-58734-4_6
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