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Domain decomposition algorithms for first-order system least squares methods. (English) Zbl 0926.65133

First-order system least squares methods have been recently proposed and analyzed for second order elliptic equations and systems. They produce symmetric and positive definite discrete systems by using standard finite element spaces which are not required to satisfy the inf-sup condition. The goal of this paper is to extend to these least squares methods some of the classical domain decomposition algorithms which have been successfully employed for standard Galerkin finite elements and to compare numerically the two approaches for simple model problems. It is shown that optimal and quasi-optimal convergence bounds follow easily from the standard Galerkin case. Numerical results confirm these bounds and show that domain decomposition algorithms for standard Galerkin and least squares discretization have comparable convergence rates. Therefore, domain decomposition provides highly parallel and scalable solvers also for first-order system least squares discretizations.
Reviewer: L.G.Vulkov (Russe)

MSC:

65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
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