# An Optimal Two-Level Overlapping Domain Decomposition Method for Elliptic Problems in Two and Three Dimensions

@article{Cai1993AnOT, title={An Optimal Two-Level Overlapping Domain Decomposition Method for Elliptic Problems in Two and Three Dimensions}, author={Xiao‐Chuan Cai}, journal={SIAM J. Sci. Comput.}, year={1993}, volume={14}, pages={239-247} }

The solution of linear systems of algebraic equations that arise from elliptic finite element problems is considered. A two-level overlapping domain decomposition method that can be viewed as a combination of the additive and multiplicative Schwarz methods is studied. This method combines the advantages of the two methods. It converges faster than the additive Schwarz algorithm and is more parallelizable than the multiplicative Schwarz algorithm, and works for general, not necessarily self…

## 49 Citations

### 1. A Family of Overlapping Schwarz Algorithms for Nonsymmetric and Indefinite Elliptic Problems

- Computer ScienceDomain-Based Parallelism and Problem Decomposition Methods in Computational Science and Engineering
- 1995

A family of overlapping Schwarz preconditioned Krylov space iterative methods for solving elliptic boundary value problems with operators that are dominated by the self adjoint second order terms but need not be eitherSelf adjoint or de nite All algorithms discussed in this paper involve two levels of preconditionsing and one of the critical components is a global coarse grid problem.

### Parallel domain decomposition preconditioning for the adaptive finite element solution of elliptic problems in three dimensions

- Computer Science
- 2001

A novel weakly overlapping two level additive Schwarz domain decomposition preconditioning algorithm is presented which is appropriate for the parallel finite element solution of elliptic partial differential equations in three dimensions and turns out to be more effective and robust than the original symmetric preconditionsing algorithm when applied to symmetric positive definite problems.

### Schwarz Methods of Neumann-Neumann Type for Three-Dimensional Elliptic Finite Element Problems

- Mathematics, Computer Science
- 1993

Several domain decomposition methods of Neumann-Neumann type are considered for solving the large linear systems of algebraic equations that arise from discretizations of elliptic problems by finite…

### Robust Iterative Methods On Unstructured Meshes

- Computer Science
- 1997

Three multilevel iterative solvers of both domain decomposition and multigrid type are proposed and analyzed, allowing almost or fully black-box implementation and suggesting the potential of the methods for solving a much wider variety of problems than those covered by the current theory.

### Iterative Solution of Elliptic Finite Element Problems on Partially Refined Meshes and the Effect of

- Computer Science
- 1993

An iterative substructuring method which is a variation of Smith's vertex space method by using the similar idea of the previous algorithm and a domain decomposition parallel preconditioning method, which is well suited for the computation of nite element solutions on multiprocessor computer architectures.

### Parallel Domain Decomposition Methods for Stochastic Elliptic Equations

- Computer ScienceSIAM J. Sci. Comput.
- 2007

It is shown theoretically and experimentally that the Schwarz preconditioned recycling GMRES method is optimal for the entire family of linear systems.

### Applications of a space decomposition method to linear and nonlinear elliptic problems

- Mathematics, Computer Science
- 1998

This work presents some space decomposition algorithms for a convex minimization problem. The algorithms has linear rate of convergence and the rate of convergence depends only on four constants. The…

### Two-Level Preconditioners For 2m'th Order Elliptic Finite Element Problems

- Computer Science
- 1995

It is shown that appropriate smoothers can be deened based on overlapping Schwarz methods and uniform preconditioning estimates are proved in the general case when the triangulations are only assumed to be shape regular but not necessarily quasiuniform.

### Domain Decomposition Preconditioning for Discontinuous Galerkin Approximations of Convection-Diffusion Problems

- Computer ScienceCSE 2009
- 2009

A class of nonoverlapping Schwarz preconditioners for DG approximations of convection-diffusion equations is studied and it is demonstrated through numerical computations that the classical Schwarz convergence theory cannot be applied to explain theoretically the converge observed numerically.

### KRYLOV CONVERGENCE ACCELERATION AND DOMAIN DECOMPOSITION METHODS FOR NONMATCHING GRIDS

- Computer Science
- 2000

This thesis develops efficient numerical solvers for partial differential equations, based on the combination of Krylov subspace methods, such as Flexible GMRES, with domain decomposition preconditioning and to extend the applicability of the developed technique to (overlapping) nonmatching grids.

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