Journal ArticleDOI
Schwarz analysis of iterative substructuring algorithms for elliptic problems in three dimensions
TLDR
Domain decomposition methods provide powerful preconditioners for the iterative solution of the large systems of algebraic equations that arise in finite element or finite difference approximations as discussed by the authors.Abstract:
Domain decomposition methods provide powerful preconditioners for the iterative solution of the large systems of algebraic equations that arise in finite element or finite difference approximations...read more
Citations
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Book ChapterDOI
Domain Decomposition Methods for Partial Differential Equations
TL;DR: This chapter summarizes basic ideas of domain decomposition methods and references are furnished to several recent uses ofdomain decomposition.
Journal ArticleDOI
Domain decomposition algorithms
Jian-Ping Shao,Tony F. Chan +1 more
TL;DR: This article surveys iterative domain decomposition techniques that have been developed in recent years for solving several kinds of partial differential equations, including elliptic, parabolic, and differential systems such as the Stokes problem and mixed formulations of elliptic problems.
Journal ArticleDOI
Convergence of Algebraic Multigrid Based on Smoothed Aggregation
TL;DR: An abstract convergence estimate is proved for the Algebraic Multigrid Method with prolongator defined by a disaggregation followed by a smoothing of the problem matrix and a matrix of the zero energy modes of the same problem but with natural boundary conditions.
Journal ArticleDOI
Dual-Primal FETI Methods for Three-dimensional Elliptic Problems with Heterogeneous Coefficients
TL;DR: It is shown that the condition numbers of several of the dual-primal FETI methods can be bounded polylogarithmically as a function of the dimension of the individual subregion problems and that the bounds are otherwise independent of the number of subdomains, the mesh size, and jumps in the coefficients.
Journal ArticleDOI
On the abstract theory of additive and multiplicative Schwarz algorithms
Michael Griebel,Peter Oswald +1 more
TL;DR: A modification of the abstract convergence theory of the additive and multiplicative Schwarz methods that makes the relation to traditional iteration methods more explicit, making convergence proofs of multilevel and domain decomposition methods clearer, or, at least, more classical.
References
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Journal ArticleDOI
GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems
Youcef Saad,Martin H. Schultz +1 more
TL;DR: An iterative method for solving linear systems, which has the property of minimizing at every step the norm of the residual vector over a Krylov subspace.
Book
The Finite Element Method for Elliptic Problems
Philippe G. Ciarlet,J. T. Oden +1 more
TL;DR: The finite element method has been applied to a variety of nonlinear problems, e.g., Elliptic boundary value problems as discussed by the authors, plate problems, and second-order problems.
Journal ArticleDOI
Iterative methods by space decomposition and subspace correction
TL;DR: A unified theory for a diverse group of iterative algorithms, such as Jacobi and Gauss–Seidel iterations, diagonal preconditioning, domain decomposition methods, multigrid methods,Multilevel nodal basis preconditionsers and hierarchical basis methods, is presented by using the notions of space decomposition and subspace correction.
BookDOI
Gesammelte mathematische Abhandlungen
TL;DR: Weierstrass et al. as discussed by the authors presented an algebraic model of the Minimalflache, which was used for the analysis of the Variationsrechnung.
Journal ArticleDOI
The construction of preconditioners for elliptic problems by substructuring. I
TL;DR: This paper develops a technique which utilizes earlier methods to derive even more efficient preconditioners for the discrete systems of equations arising from the numerical approximation of elliptic boundary value problems.