A note on the complexity of solving Poisson's equation for spaces of bounded mixed derivatives
type: Item
instance of: scholarly article
title: A note on the complexity of solving Poisson's equation for spaces of bounded mixed derivatives
zbMATH Open document ID: 0954.65078
published in: Journal of Complexity
publication date: 2000-05-07T00:00:00Z
review text: This paper studies the solution of the Poisson equation in a \(d\)-dimensional cube using a hierarchical, tensor product finite element basis on sparse grids. Whereas on earlier sparse grids the number of unknowns (and operations) was \(O(h^{-1}|\log h|^{d-1})\) guaranteeing an accuracy of \(O(h)\), now an even sparser grid is proposed (thanks to an explicit solution of a corresponding optimization problem) leading to a discrete problem with \(O(h^{-1})\) unknowns which can be solved with a multilevel preconditioner based on prewavelets [constructed by \textit{M. Griebel} and \textit{P. Oswald}, Adv. Comput. Math. 4, No.~1-2, 171-206 (1995; Zbl 0826.65099)] in \(O(h^{-1})\) operations. This approach gives an accuracy of \(O(h)\) in the energy norm, assuming the solution of the original continuous problem to have bounded mixed derivatives (in the \(L_\infty\)- or \(L_2\)-norm) up to order \(2d\).
reviewed by: Gisbert Stoyan
Mathematics Subject Classification ID: 65N30
Mathematics Subject Classification ID: 65F10
Mathematics Subject Classification ID: 65F35
Mathematics Subject Classification ID: 65N15
Mathematics Subject Classification ID: 65N55
Mathematics Subject Classification ID: 35J05
zbMATH DE Number: 1439899
zbMATH Keywords: sparse grids
zbMATH Keywords: hierarchical finite elements
zbMATH Keywords: Poisson equation
zbMATH Keywords: error bounds
zbMATH Keywords: multilevel preconditioner
zbMATH Keywords: prewavelets
author: Hans-Joachim Bungartz
author: Michael Griebel
MaRDI profile type: Publication
OpenAlex ID: W2052806039
cites work: Q3284729
cites work: Q4328438
cites work: Parallel Multilevel Preconditioners
cites work: Q4742248
cites work: Q3140820
cites work: Q4246983
cites work: Multilevel preconditioning
cites work: d-Variate Boolean interpolation
cites work: Revenge of the Semicoarsening Frequency Decomposition Multigrid Method
cites work: Grandchild of the Frequency Decomposition Multigrid Method
cites work: Q5686440
cites work: Q4010700
cites work: Q4840187
cites work: Q4338777
cites work: Optimized general sparse grid approximation spaces for operator equations
cites work: On additive Schwarz preconditioners for sparse grid discretizations
cites work: Tensor product type subspace splittings and multilevel iterative methods for anisotropic problems
cites work: Q3140818
cites work: Multilevel Gauss-Seidel-algorithms for full and sparse grid problems
cites work: Q3750065
cites work: Q3698230
cites work: The frequency decomposition multi-grid method. I: Application to anisotropic equations
cites work: The frequency decomposition multi-grid method. II: Convergence analysis based on the additive Schwarz method
cites work: Q3123963
cites work: Sparse-grid finite-volume multigrid for \(3D\)-problems
cites work: On theh- andp-Versions of the Extrapolated Gordon’s Projector with Applications to Elliptic Equations
cites work: On the design of nested iterations for elliptic difference equations
cites work: Q3993418
cites work: A new multigrid approach to convection problems
cites work: Q4840194
cites work: Q4877220
cites work: Q5600030
cites work: Q4404383
cites work: On approximate recovery of functions with bounded mixed derivative
cites work: Q3993279
cites work: Explicit cost bounds of algorithms for multivariate tensor product problems
cites work: On the multi-level splitting of finite element spaces
cites work: Hierarchical bases of finite-element spaces in the discretization of nonsymmetric elliptic boundary value problems
cites work: Q4010713
DOI: 10.1006/JCOM.1999.0499
description: scientific article; zbMATH DE number 1439899