## Overlapping Schwarz domain decomposition preconditioners for the local discontinuous Galerkin method for elliptic problems.(English)Zbl 1232.65164

At first the local discontinuous Galerkin discretization of a Poisson problem is described. Then, two-level overlapping additive Schwarz preconditioners are introduced. Estimates of the condition number of the preconditioned system are established. Hereby, two cases are considered, namely general overlaps and small overlaps. It is shown that the condition number behaves like $$1 + H^2/\delta^2$$ in the case of general overlap, where $$H$$ is the coarse mesh size and $$\delta$$ measures the overlap. Under additional conditions on the subdomains the improved condition number bound $$C(1 + H/\delta)$$ is proved, where the constant $$C$$ is independent of $$H$$, $$\delta$$, the fine mesh size, and the number of subdomains. Finally, numerical experiments are presented which confirm the theoretical results and show the good parallel scalability of the proposed preconditioners.

### 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 65Y05 Parallel numerical computation 35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation 65F08 Preconditioners for iterative methods

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