A note on the complexity of solving Poisson's equation for spaces of bounded mixed derivatives

DOI10.1006/jcom.1999.0499zbMath0954.65078OpenAlexW2052806039MaRDI QIDQ1974571

Michael Griebel, Hans-Joachim Bungartz

Publication date: 7 May 2000

Published in: Journal of Complexity (Search for Journal in Brave)

Full work available at URL: https://semanticscholar.org/paper/24fdb1ff442eb347d3629d4b955aba1f04165430

zbMATH Keywords

error boundssparse gridsPoisson equationprewaveletsmultilevel preconditionerhierarchical finite elements

Mathematics Subject Classification ID

Multigrid methods; domain decomposition for boundary value problems involving PDEs (65N55) Error bounds for boundary value problems involving PDEs (65N15) Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs (65N30) Iterative numerical methods for linear systems (65F10) Numerical computation of matrix norms, conditioning, scaling (65F35) Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation (35J05)

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Cites Work

• Hierarchical bases of finite-element spaces in the discretization of nonsymmetric elliptic boundary value problems
• On the multi-level splitting of finite element spaces
• A new multigrid approach to convection problems
• The frequency decomposition multi-grid method. I: Application to anisotropic equations
• Multilevel preconditioning
• The frequency decomposition multi-grid method. II: Convergence analysis based on the additive Schwarz method
• On additive Schwarz preconditioners for sparse grid discretizations
• Explicit cost bounds of algorithms for multivariate tensor product problems
• d-Variate Boolean interpolation
• Sparse-grid finite-volume multigrid for \(3D\)-problems
• Tensor product type subspace splittings and multilevel iterative methods for anisotropic problems
• On approximate recovery of functions with bounded mixed derivative
• Multilevel Gauss-Seidel-algorithms for full and sparse grid problems
• Optimized general sparse grid approximation spaces for operator equations
• Parallel Multilevel Preconditioners
• On theh- andp-Versions of the Extrapolated Gordon’s Projector with Applications to Elliptic Equations
• Revenge of the Semicoarsening Frequency Decomposition Multigrid Method
• Grandchild of the Frequency Decomposition Multigrid Method
• On the design of nested iterations for elliptic difference equations
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