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Two-level additive Schwarz methods for a discontinuous Galerkin approximation of second order elliptic problems. (English) Zbl 1007.65104
The finite element functions in the framework of discontinuous Galerkin methods are not continuous at interelement boundaries. This is compensated by two modifications of the energy functional. A penalty term for the jumps is added, and another term corresponds to the terms with Lagrange multipliers, and the latter are replaced by normal derivatives. Now it is easier to apply domain decomposition due to the missing continuity assumptions. Some trace theorems are derived that can replace the Poincaré-Friedrichs inequality in this framework. As usual, condition numbers $$O(H/h)$$ for nonoverlapping Schwarz methods and $$O(H/\delta)$$ for the overlapping case, respectivly, are derived.

MSC:
 65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs 65F10 Iterative numerical methods for linear systems
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