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**Optimal iterative refinement methods.**
*(English)*
Zbl 0682.65066

Domain decomposition methods, Proc. 2nd Int. Symp., Los Angeles/Calif. 1988, 114-125 (1989).

[For the entire collection see Zbl 0675.00021.]

Author’s summary: We consider the solution of the linear systems of algebraic equations which arise from elliptic finite element problems defined on composite meshes. Such problems can systematically be built up by introducing a basic finite element approximation on the entire region and then repeatedly selecting subregions, and subregions of subregions, where the finite element model is further refined in order to gain higher accuracy. We consider conjugate gradient algorithms, and other acceleration procedures, where, in each iteration, problems representing finite element models on the original region and the subregions prior to further refinement are solved. We can therefore use solvers for problems with uniform or relatively uniform mesh sizes, while the composite mesh can be strongly graded.

In this contribution to the theory, we report on new results recently obtained in joint work with M. Dryja [An Additive Variant of the Schwarz Alternating Method for the Case of Many Subregions, Technical Report 339; also Ultracomputer Note 131, Dept. of Computer Science, Courant Institute (1987)]. We use a basic mathematical framework recently introduced in a study of a variant of Schwarz’ alternating algorithm. We establish that several fast methods can be devised which are optimal in the sense that the number of iterations required to reach a certain tolerance is independent of the mesh size as well as the number of refinement levels. This work is also technically quite closely related to previous work on iterative substructuring methods, which are domain decomposition algorithms using non-overlapping subregions.

Author’s summary: We consider the solution of the linear systems of algebraic equations which arise from elliptic finite element problems defined on composite meshes. Such problems can systematically be built up by introducing a basic finite element approximation on the entire region and then repeatedly selecting subregions, and subregions of subregions, where the finite element model is further refined in order to gain higher accuracy. We consider conjugate gradient algorithms, and other acceleration procedures, where, in each iteration, problems representing finite element models on the original region and the subregions prior to further refinement are solved. We can therefore use solvers for problems with uniform or relatively uniform mesh sizes, while the composite mesh can be strongly graded.

In this contribution to the theory, we report on new results recently obtained in joint work with M. Dryja [An Additive Variant of the Schwarz Alternating Method for the Case of Many Subregions, Technical Report 339; also Ultracomputer Note 131, Dept. of Computer Science, Courant Institute (1987)]. We use a basic mathematical framework recently introduced in a study of a variant of Schwarz’ alternating algorithm. We establish that several fast methods can be devised which are optimal in the sense that the number of iterations required to reach a certain tolerance is independent of the mesh size as well as the number of refinement levels. This work is also technically quite closely related to previous work on iterative substructuring methods, which are domain decomposition algorithms using non-overlapping subregions.

Reviewer: S.F.McCormick

### MSC:

65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |

65F10 | Iterative numerical methods for linear systems |

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