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Variational formulation and algorithm for trace operator in domain decomposition calculations. (English) Zbl 0684.65094

Domain decomposition methods, Proc. 2nd Int. Symp., Los Angeles/Calif. 1988, 3-16 (1989).
[For the entire collection see Zbl 0675.00021.]
The authors propose a new preconditioning strategy for solving second order elliptic boundary value problems (mixed Dirichlet and Neumann) via domain composition techniques. The preconditioning is done by introducing an operator of Steklov-Poincarés type which is defined on the product of subdomain spaces. The operator involves the solution of a Dirichlet problem on each subdomain, trace averaging and the solution of a Neumann problem on each subdomain. The problem can then be written in variational form; it is solved iteratively by a conjugate gradient algorithm.
Two- and three-dimensional numerical results are given. They show that the algorithm can operate on arbitrary geometries and unstructured meshes. Convergence is independent of the initial trace value and the discretization step.
Reviewer: J.Weisel

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations

Citations:

Zbl 0675.00021