# Parallel multilevel preconditioners

@article{Bramble1990ParallelMP, title={Parallel multilevel preconditioners}, author={James H. Bramble and Joseph E. Pasciak and Jinchao Xu}, journal={Mathematics of Computation}, year={1990}, volume={55}, pages={1-22} }

In this paper, we shall report on some techniques for the development of preconditioners for the discrete systems which arise in the approximation of solutions to elliptic boundary value problems. Here we shall only state the resulting theorems. It has been demonstrated that preconditioned iteration techniques often lead to the most computationally effective algorithms for the solution of the large algebraic systems corresponding to boundary value problems in two and three dimensional Euclideanâ€¦Â

## 699 Citations

Domain Decomposition Methods for Problems with Partial Refinement

- Computer Science, MathematicsSIAM J. Sci. Comput.
- 1992

It is proven that the iteration schemes converge to the discrete solution at a rate which is independent of the mesh parameters in the case of two spatial dimensions, and for the iterative convergence rate in three dimensions are somewhat weaker.

Robust Parallel Newton { Multilevel

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The present paper is devoted to the numerical solution of nonlinear boundary value problems arising in the magnetic eld computation and in solid mechanics, and compares by numerical examples the performance of the diierent iterative solvers which are applied within each Newton step.

Continuous preconditioners for the mixed Poisson problem

- Computer Science, Mathematics
- 2012

The preconditioner can be applied both to the original and to the reduced Schur complement problem, and the number of iterations required to solve the preconditionsed system will have the same dependency on the mesh size as for the precondo applied to the Poisson problem.

Modified hierarchy basis for solving singular boundary value problems

- Computer Science, Mathematics
- 2007

Parallel Multi-level Solvers for Elliptic Boundary Value Problems in Three-Dimensional Domains

- Computer Science
- 1998

The algorithms presented are applied to solve linear systems of finite element equations arising from the discretization of elliptic boundary value problems in three-dimensional domains, where tetrahedral elements with piecewise linear or piecewise quadratic functions are used.

The Combination Technique for Parallel Sparse-Grid-Preconditioning or -Solution of PDEs on Workstation Networks

- Computer ScienceCONPAR
- 1992

This paper studies the parallel solution of elliptic partial differential equations with the sparse grid combination technique and describes the resulting algorithm, which can be used as a solver and within a preconditioner.

Field-of-values analysis of preconditioned iterative methods for nonsymmetric elliptic problems

- Computer Science
- 1997

Bounds on the convergence rate of Krylov subspace methods for the solution of nonsymmetric systems of linear equations, such as GMRES or FOM, are presented which are based on the smallest real part of the field of values of the coefficient matrix and of its inverse.

Preconditioning for boundary element methods in domain decomposition

- Mathematics, Computer Science
- 2001

Preconditioning a class of fourth order problems by operator splitting

- Computer Science, MathematicsNumerische Mathematik
- 2011

This work proposes symmetric and non-symmetric preconditioners for systems arising from finite element discretizations of parabolic problems which are fourth order in space, and considers boundary conditions which yield a natural splitting of the discretized fourth order operator into two linear second order elliptic operators.

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