The Role of the Strengthened Cauchy–Buniakowskii–Schwarz Inequality in Multilevel Methods

DOI10.1137/1033098zbMath0737.65026OpenAlexW2083054768MaRDI QIDQ3981403

Panayot S. Vassilevski, Victor Eijkhout

Publication date: 26 June 1992

Published in: SIAM Review (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1137/1033098

zbMATH Keywords

Schur complementfinite element methodssurvey paperelliptic problemsmultilevel methodsmultilevel preconditionerspreconditioned iterative methodsstrengthened Cauchy-Buniakowskii-Schwarz inequality

Mathematics Subject Classification ID

Boundary value problems for second-order elliptic equations (35J25) 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) Numerical solution of discretized equations for boundary value problems involving PDEs (65N22)

Related Items

Anisotropic finite elements for the Stokes problem: a posteriori error estimator and adaptive mesh, Error Estimation and Adaptivity for Stochastic Collocation Finite Elements Part I: Single-Level Approximation, A Posteriori Finite Element Error Analysis for Symmetric Positive Differential Equations, CBS constants \& their role in error estimation for stochastic Galerkin finite element methods, A New Analytical Framework for the Convergence of Inexact Two-Grid Methods, Weak residual error estimates for symmetric positive systems, A posteriori estimates using auxiliary subspace techniques, Semi-coarsening AMLI preconditioning of anisotropic trilinear FEM systems, A posteriori finite element error estimators for indefinite elliptic boundary value problems^∗, Rapid error reduction for block Gauss-Seidel based onp-hierarchical basis, Properties of the multidimensional finite elements, A simple analysis of some a posteriori error estimates, A posteriori estimates of Taylor-Hood element for Stokes problem using auxiliary subspace techniques, An adaptive weak Galerkin finite element method with hierarchical bases for the elliptic problem, An a posteriori estimator of eigenvalue/eigenvector error for penalty-type discontinuous Galerkin methods, On the convergence of an optimal additive Schwarz method for parallel adaptive finite elements, New variants of the algebraic multilevel preconditioning methods, A posteriori error estimation and adaptivity in stochastic Galerkin FEM for parametric elliptic PDEs: beyond the affine case, A posteriori error estimator for weak Galerkin finite element method for Stokes problem using diagonalization techniques, On element-by-element Schur complement approximations, Error Analysis of Supremizer Pressure Recovery for POD based Reduced-Order Models of the Time-Dependent Navier--Stokes Equations, An optimal preconditioned FFT-accelerated finite element solver for homogenization, On hybrid v-cycle version (i) algebraic multilevel preconditionings, Robust AMLI methods for parabolic Crouzeix-Raviart FEM systems, Algebraic multilevel preconditioning in isogeometric analysis: construction and numerical studies, A hierarchical iterative procedure for the analysis of composite laminates, An a posteriori finite element error analysis for the Stokes equations, Robust a posteriori error estimation for parameter-dependent linear elasticity equations, Efficient Adaptive Algorithms for Elliptic PDEs with Random Data, CBS constants for multilevel splitting of graph-Laplacian and application to preconditioning of discontinuous Galerkin systems, Multilevel preconditioning of rotated bilinear non-conforming FEM problems, Effective condition number of Trefftz methods for biharmonic equations with crack singularities, A Posteriori Error Analysis of Any Order Finite Volume Methods for Elliptic Problems, Multilevel algorithms for Rannacher-Turek finite element approximation of 3D elliptic problems, Efficient Adaptive Stochastic Galerkin Methods for Parametric Operator Equations, Strengthened Cauchy-Schwarz inequality for biorthogonal wavelets in Sobolev spaces, A convergence theory for an overlapping Schwarz algorithm using discontinuous iterates, Guaranteed lower eigenvalue bounds for two spectral problems arising in fluid mechanics, Decomposition into subspaces preconditioning: abstract framework, Block Preconditioning of Stochastic Galerkin Problems: New Two-sided Guaranteed Spectral Bounds, Strengthened Cauchy-Bunyakowski-Schwarz inequality for a three-dimensional elasticity system, An Artificial Compression Reduced Order Model, Discretization of Linear Problems in Banach Spaces: Residual Minimization, Nonlinear Petrov--Galerkin, and Monotone Mixed Methods, A posteriori finite element error estimators for parametrized nonlinear boundary value problems, Algebraic analysis of two-grid methods: The nonsymmetric case, A Bauer-Hausdorff matrix inequality, A construction of higher-order finite volume methods, An improved multigrid solver for the \(p\)-hierarchical basis finite element method using a space decomposition smoother, A posteriori boundary element error estimation, Finite element approximation of multi-scale elliptic problems using patches of elements, Restrictive preconditioners for conjugate gradient methods for symmetric positive definite linear systems, An optimal Schwarz preconditioner for a class of parallel adaptive finite elements, Algebraic multilevel iteration method for lowest order Raviart-Thomas space and applications