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Iterative Methods by Space Decomposition and Subspace Correction
  • Jinchao Xu
  • Computer Science
    SIAM Rev.
  • 1 December 1992
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.
Parallel multilevel preconditioners
This paper discusses an approach for developing completely parallel multilevel preconditioners and describes the simplest application of the technique to a model elliptic problem.
Two-grid Discretization Techniques for Linear and Nonlinear PDEs
A number of finite element discretization techniques based on two (or more) subspaces for nonlinear elliptic partial differential equations (PDEs) is presented and optimal error estimates are obtained.
Nodal Auxiliary Space Preconditioning in H(curl) and H(div) Spaces
This paper develops and analyzes a general approach to preconditioning linear systems of equations arising from conforming finite element discretizations of H(curl, )- and H(div,)-elliptic variational problems and proves mesh-independent effectivity of the precondITIONers by using the abstract theory of auxiliary space preconditionsing.
The analysis of multigrid algorithms with nonnested spaces or noninherited quadratic forms
A theory for the analysis of multigrid algorithms for symmetric positive definite problems with nonnested spaces and noninherited quadratic forms is provided and various numerical approximations of second-order elliptic boundary value problems are applied.
A two-grid discretization scheme for eigenvalue problems
A two-grid discretization scheme is proposed for solving eigenvalue problems, including both partial differential equations and integral equations, and maintains an asymptotically optimal accuracy.
Local and parallel finite element algorithms based on two-grid discretizations
A number of new local and parallel discretization and adaptive nite element algorithms are proposed and analyzed in this paper for elliptic boundary value problems and the main idea is to use a coarse grid to approximate the low frequencies and then to correct the resulted residue (which contains mostly high frequencies) by some local/parallel procedures.
Convergence estimates for multigrid algorithms without regularity assumptions
A new technique for proving rate of convergence estimates of multi- grid algorithms for asymmetric positive definite problems for symmetricpositive definite problems will be given in this paper.
The method of alternating projections and the method of subspace corrections in Hilbert space
The method of alternating projections and the method of subspace corrections are general iterative methods that have a variety of applications. The method of alternating projections, first proposed
Asymptotically Exact A Posteriori Error Estimators, Part I: Grids with Superconvergence
This work develops superconvergence estimates for piecewise linear finite element approximations on quasi-uniform triangular meshes where most pairs of triangles sharing a common edge form approximate parallelograms and analyses a postprocessing gradient recovery scheme for general unstructured, shape regular triangulations.