Abstract
We present some two-level non-overlapping additive and multiplicative Schwarz methods for a discontinuous Galerkin method for solving the biharmonic equation. We show that the condition numbers of the preconditioned systems are of the order O( H3/h3) for the non-overlapping Schwarz methods, where h and H stand for the fine mesh size and the coarse mesh size, respectively. The analysis requires establishing an interpolation result for Sobolev norms and Poincare--Friedrichs type inequalities for totally discontinuous piecewise polynomial functions. It also requires showing some approximation properties of the multilevel hierarchy of discontinuous Galerkin finite element spaces.
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