Two-Level Non-Overlapping Schwarz Preconditioners for a Discontinuous Galerkin Approximation of the Biharmonic Equation

Authors:
Xiaobing Feng;Ohannes A. Karakashian
Affiliations:
Department of Mathematics, University of Tennessee, Knoxville, USA 37996;Department of Mathematics, University of Tennessee, Knoxville, USA 37996
Venue:
Journal of Scientific Computing
Year:
2005

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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 Poincaré--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.