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On the Representation of Functions and Finite Difference Operators on Adaptive Dyadic Grids

  • In this paper we describe methods to approximate functions and differential operators on adaptive sparse (dyadic) grids. We distinguish between several representations of a function on the sparse grid and we describe how finite difference (FD) operators can be applied to these representations. For general variable coefficient equations on sparse grids, genuine finite element (FE) discretizations are not feasible and FD operators allow an easier operator evaluation than the adapted FE operators. However, the structure of the FD operators is complex. With the aim to construct an efficient multigrid procedure, we analyze the structure of the discrete Laplacian in its hierarchical representation and show the relation between the full and the sparse grid case. The rather complex relations, that are expressed by scaling matrices for each separate coordinate direction, make us doubt about the possibility of constructing efficient preconditioners that show spectral equivalence. Hence, we question the possibility of constructing a natural multigrid algorithm with optimal O(N) efficiency. We conjecture that for the efficient solution of a general class of adaptive grid problems it is better to accept an additional condition for the dyadic grids (condition L) and to apply adaptive hp-discretization.

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Author:P.W. Hemker, Frauke SprengelORCiDGND
DOI original:https://doi.org/10.2478/cmam-2001-0016
Parent Title (English):Computational Methods in Applied Mathematics
Document Type:Article
Year of Completion:2001
Publishing Institution:Hochschule Hannover
Release Date:2023/11/23
Tag:Dyadisches Gitter
adaptive methods; dyadic grid; sparse grid
GND Keyword:Dünnes Gitter; Adaptives Verfahren
Page Number:20
First Page:222
Last Page:242
Link to catalogue:1871109477
Institutes:Fakultät IV - Wirtschaft und Informatik
DDC classes:510 Mathematik
Licence (German):License LogoCreative Commons - CC BY-NC-ND - Namensnennung - Nicht kommerziell - Keine Bearbeitungen 4.0 International