Abstract
Efficient discretization techniques are of crucial importance for most types of problems in numerical mathematics, starting from tasks like how to define sets of points to approximate, interpolate, or integrate certain classes of functions as good as possible, up to the numerical solution of differential equations. Introduced by Zenger in 1990 and based on hierarchical tensor product approximation spaces, sparse grids have turned out to be a very efficient approach in order to improve the ratio of invested storage and computing time to the achieved accuracy for many problems in the areas mentioned above.
In this paper, we discuss two new algorithmic developments concerning the sparse grid finite element discretization of elliptic partial differential equations. First, a method for the numerical treatment of the general linear elliptic differential operator of second order is presented which, with the help of mapping techniques, allows to tackle problems on more complicated geometries. Second, we leave the approximation space of the piecewise multilinear functions and introduce hierarchical polynomial bases of piecewise arbitrary degree that lead to a very straightforward and efficient access to an approximation of higher order on sparse grids.
Both algorithms discussed here have been designed in a unidirectional way that allows the recursive reduction of the general d-dimensional case to the simpler 1D one and, thus, the formulation of programs for arbitrary d.
This work has been supported by the Bayerische Forschungsstiftung via FORTWIHR — The Bavarian Consortium for HPSC.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Amsden, A. A., Hirt, C. W.: A simple scheme for generating general curvilinear grids. J. Comp. Phys. 11 (1973) 348–359
Babuška, I., Suri, M.: The p-and h-p-versions of the finite element method: An overview. Comput. Meth. Appl. Mech. Engrg. 80 (1990) 5–26
Babuska, L, Szabó, B.A., Katz, I.N.: The p-version of the finite element method. SIAM J. Num. Anal. 18 (1981) 515–545
Balder, R., Zenger, C: The solution of multidimensional real Helmholtz equations on sparse grids. SIAM J. Sci.; Comp. 17 (1996) 631–646
Bank, R.E., Dupont, T., Yserentant, H.: The hierarchical basis multigrid method. Num. Math. 52 (1988) 427–458
Bonk, T.: A new algorithm for multi-dimensional adaptive numerical quadrature. In Hackbusch, W., Wittum, G.: Adaptive Methods: Algorithms, Theory, and Applications. NNFM 46 Vieweg Braunschweig (1994)
Bungartz, H.-J.: An adaptive Poisson solver using hierarchical bases and sparse grids. In de Groen, P., Beauwens, R.: Iterative Methods in Linear Algebra. Elsevier Amsterdam (1992) 293–310
Bungartz, H.-J.: Dünne Gitter und deren Anwendung bei der adaptiven Lösung der dreidimensionalen Poisson-Gleichung. Dissertation, Institut für Informatik, TU München (1992)
Bungartz, H.-J.: Concepts for higher order finite elements on sparse grids. In Ilin, A. V., Scott, L. R.: Proc. 3rd Int. Conf. on Spectral and High Order Methods. Houston Journal of Mathematics (1996) 159–170
Bungartz, H.-J., Griebel, M., Röschke, D., Zenger, C.: Pointwise convergence of the combination technique for Laplace’s equation. East-West J. Num. Math. 2 (1994) 21–45
Bungartz, H.-J., Griebel, M., Rüde, U.: Extrapolation, combination, and sparse grid techniques for elliptic boundary value problems. Comput. Meth. Appl. Mech. Engrg. 116 (1994) 243–252
Dornseifer, T., Pflaum, C.: Discretization of elliptic partial differential equations on curvilinear bounded domains with sparse grids. Computing 56 (1996) 197–214
Gordon, W. J., Hall, C. A.: Construction of curvilinear coordinate systems and application to mesh generation. Int. J. Num. Meth. Eng. 7 (1973) 461–477
Gordon, W. J., Thiel, C: Transfinite mappings and their application to grid generation. In Thompson, J.F.: Numerical Grid Generation. North-Holland Amsterdam (1982) 171–192
Griebel, M.: Parallel multigrid methods on sparse grids. In Hackbusch, W., Trottenberg, U.: Multigrid Methods III. Int. Ser. Num. Math. 3 Birkhäuser Basel (1991) 211–221
Griebel, M.: A parallelizable and vectorizable multi-level algorithm on sparse grids. In Hackbusch, W.: Parallel Algorithms for Partial Differential Equations. NNFM 31 Vieweg Braunschweig (1991) 94–100
Griebel, M., Oswald, P.: On additive Schwarz preconditioners for sparse grid discretizations. Num. Math. 66 (1994) 449–464
Griebel, M., Schneider, M., Zenger, C: A combination technique for the solution of sparse grid problems. In de Groen, P., Beauwens, R.: Iterative Methods in Linear Algebra. Elsevier Amsterdam (1992) 263–281
Guo, B., Babuska, I.: The h-p-version of the finite element method (Part 1: The basic approximation results). Comput. Mech. 1 (1986) 21–41
Guo, B., Babuska, L: The h-p-version of the finite element method (Part 2: General results and applications). Comput. Mech. 1 (1986) 203–220
Hallatschek, K.: Fouriertransformation auf dünnen Gittern mit hierarchischen Basen. Num. Math. 63 (1992) 83–97
Hemker, P. W.: Sparse grid finite-volume multigrid for 3D problems. Advances in Comput. Mech. (Special issue on multiscale problems) (1994) 215–232
Hilgenfeldt, S.: Numerische Lösung der stationären Schrödingergleichung mit Finite-Element-Methoden auf dünnen Gittern. Diplomarbeit, Institut für Informatik, TU München (1994)
Knupp, P., Steinberg, S.: Fundamentals of Grid Generation. CRC Press London (1993)
Liao, G.: On harmonic maps. In Castillo, J. E.: Mathematical Aspects of Numerical Grid Generation. SIAM Philadelphia (1991)
Pflaum, C: Diskretisierung elliptischer Differentialgleichungen mit dünnen Gittern. Dissertation, Institut für Informatik, TU München (1996)
Störtkuhl, T.: Ein numerisches, adaptives Verfahren zur Lö sung der biharmonischen Gleichung auf dünnen Gittern. Dissertation, Institut für Informatik, TU München (1994)
Thompson, J. F., Thames, F. C, Mastin, C.W.: Automatic numerical generation of boundary-fitted curvilinear coordinate system for field containing any number of arbitrary two-dimensional bodies. J. Comp. Phys. 15 (1974) 299–319
Yserentant, EL: On the multilevel splitting of finite element spaces. Num. Math. 49 (1986) 379–412
Zenger, C: Sparse grids. In Hackbusch, W.: Parallel Algorithms for Partial Differential Equations. NNFM 31 Vieweg Braunschweig (1991)
Zumbusch, G.W.: Simultaneous h-p Adaption in Multilevel Finite Elements. Shaker Aachen (1996)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1998 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Bungartz, HJ., Dornseifer, T. (1998). Sparse Grids: Recent Developments for Elliptic Partial Differential Equations. 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_3
Download citation
DOI: https://doi.org/10.1007/978-3-642-58734-4_3
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-63133-0
Online ISBN: 978-3-642-58734-4
eBook Packages: Springer Book Archive