An additive Schwarz method for variational inequalities. (English) Zbl 0954.65051
The objective of this paper is to exploit a convergence theory for the additive Schwarz method, for variational inequalities and their approximations by finite element methods. Main result: The Schwarz domain decomposition method is proved to converge with a geometric rate depending on the decomposition of the domain. The result is based on an abstract framework of convergence analysis established for general variational inequalities in Hilbert spaces.
The theory for the obstacle problem is demonstrated. Finally, the Schwarz method in the finite element spaces is analyzed.
The theory for the obstacle problem is demonstrated. Finally, the Schwarz method in the finite element spaces is analyzed.
Reviewer: J.Lovíšek (Bratislava)
MSC:
| 65K10 | Numerical optimization and variational techniques |
| 49M27 | Decomposition methods |
| 49J40 | Variational inequalities |
| 49M15 | Newton-type methods |
| 65N55 | Multigrid methods; domain decomposition for boundary value problems involving PDEs |
| 35J85 | Unilateral problems; variational inequalities (elliptic type) (MSC2000) |
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
Keywords:
variational inequalities; obstacle problems; finite element methods; domain decomposition methods; convergence; additive Schwarz methodReferences:
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