Abstract
We propose some algorithms to solve the system of linear equations arising from the finite difference discretization on sparse grids. For this, we will use the multilevel structure of the sparse grid space or its full grid subspaces, respectively.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
H.-J. Bungartz and T. Dornseifer, Sparse grids: Recent developments for elliptic partial differential equations, in: Multigrid Methods V, eds. W. Hackbusch and G. Wittum, Lecture Notes in Computational Science and Engineering, Vol 3 (Springer, Berlin, 1998).
F.-J. Delvos and H. Posdorf, Nth order blending, in: Constructive Theory of Functions of Several Variables, eds. W. Schempp and K. Zeller (Springer, Berlin, 1977) pp. 53–64.
F.-J. Delvos and W. Schempp, BooleanMethods in Interpolation and Approximation, Pitman Research Notes in Mathematics Series, Vol. 230 (Longman Scientific & Technical, Harlow, 1989).
M. Griebel, Multilevelmethoden als Iterationsverfahren über Erzeugendensystemen, Teubner Skripten zur Numerik (Teubner, Stuttgart, 1994).
M. Griebel, Adaptive sparse grid multilevel methods for elliptic PDEs based on finite differences, Computing 61 (1998) 151–179.
M. Griebel and P. Oswald, Tensor product type subspace splitting and multilevel iterative methods for anisotropic problems, Adv. Comput. Math. 4 (1995) 171–206.
M. Griebel, M. Schneider and C. Zenger, A combination technique for the solution of sparse grid problems, in: Iterative Methods in Linear Algebra, eds. R. Beauwens and P. de Groen, IMACS (Elsevier, Amsterdam, 1992) pp. 263–281.
P. W. Hemker, Sparse-grid finite-volume multigrid for 3D-problems, Adv. Comput. Math. 4 (1995) 83–110.
P. W. Hemker and C. Pflaum, Approximation on partially ordered sets of regular grids, Appl. Numer. Math. 25 (1997) 55–87.
P.W. Hemker and F. Sprengel, On the representation of functions and finite difference operators on adaptive dyadic grids (2000) submitted (CWI Report MAS-R9933).
P.W. Hemker and F. Sprengel, Experience with the solution of a finite difference discretization on sparse grids (2000) submitted (GMD report 98).
N.M. Korobov, Approximate calculation of multiple integrals with the aid of methods in the theory of numbers, Dokl. Akad. Nauk SSSR 115 (1957) 1062–1065 (in Russian).
J. Noordmans and P.W. Hemker, Convergence results for 3D sparse grid approaches, Numer. Linear Algebra Appl. 5 (1999) 363–376.
P. Oswald, Multilevel Finite Element Approximation. Theory and Applications, Teubner Skripten zur Numerik (Teubner, Stuttgart, 1994).
T. Schiekofer, Die Methode der Finiten Differenzen auf dünnen Gittern zur Lösung elliptischer und parabolischer partieller Differentialgleichungen, Ph.D. thesis, Universität Bonn (1998).
S.A. Smolyak, Quadrature and interpolation formulas the classes W a s and E a s Dokl. Akad. Nauk SSSR. 1 (1963) 384–387.
F. Sprengel, Interpolation and wavelet decomposition of multivariate periodic functions, Ph.D. thesis, University of Rostock (1997).
V.N. Temlyakov, Approximation of Periodic Functions (Nova Science, New York, 1993).
C. Zenger, Sparse grids, in: Parallel Algorithms for Partial Differential Equations, ed. W. Hackbusch, Notes on Numerical Fluid Mechanics, Vol. 31 (Vieweg, Braunschweig, 1991) pp. 297–301.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Sprengel, F. Multilevel algorithms for finite difference discretizations on sparse grids. Numerical Algorithms 26, 111–121 (2001). https://doi.org/10.1023/A:1016651510463
Issue Date:
DOI: https://doi.org/10.1023/A:1016651510463