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**Local defect correction method and domain decomposition techniques.**
*(English)*
Zbl 0552.65070

Defect correction methods. Theory and applications, Comput. Suppl. 5, 89-113 (1984).

[For the entire collection see Zbl 0545.00019.]

Author’s summary: For elliptic problems a local defect correction method is described. A basic (global) discretization is improved by a local discretization defined in a subdomain. The convergence rate of the local defect correction iteration is proved to be proportional to a certain positive power of the step size. The accuracy of the converged solution can be described. Numerical examples confirm the theoretical results. We discuss multi-grid iterations converging to the same solution. The local defect correction determines a solution depending on one global and one or more local discretizations. An extension of this approach is the domain decomposition method, where only (overlapping) local problems are combined. Such a combination of local subproblems can be solved efficiently by a multi-grid iteration. We describe a multi-grid variant that is suited for the use of parallel processors.

Author’s summary: For elliptic problems a local defect correction method is described. A basic (global) discretization is improved by a local discretization defined in a subdomain. The convergence rate of the local defect correction iteration is proved to be proportional to a certain positive power of the step size. The accuracy of the converged solution can be described. Numerical examples confirm the theoretical results. We discuss multi-grid iterations converging to the same solution. The local defect correction determines a solution depending on one global and one or more local discretizations. An extension of this approach is the domain decomposition method, where only (overlapping) local problems are combined. Such a combination of local subproblems can be solved efficiently by a multi-grid iteration. We describe a multi-grid variant that is suited for the use of parallel processors.

Reviewer: J.Mandel

### MSC:

65N22 | Numerical solution of discretized equations for boundary value problems involving PDEs |

65N50 | Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs |

35J25 | Boundary value problems for second-order elliptic equations |