Abstract
We present some two-level non-overlapping and overlapping additive Schwarz methods for discontinuous Galerkin methods for the biharmonic equation. It is shown that the condition numbers of the preconditioned systems are of the order \(O\left( \left( \frac{H}{h}\right) ^3\right) \) for the non-overlapping method, and of the order \(O\left( \frac{H}{h}+\left( \frac{H}{\delta }\right) ^3\right) \) for the overlapping method, where h and H stand for the fine and coarse mesh sizes respectively, and \(\delta \) denotes the size of the overlaps between subdomains. In particular, emphasis is placed on studying the influence of the penalty parameters and the choice of the coarse mesh bilinear form on the condition numbers. Numerical experiments are provided to gauge the efficiency of the methods and to validate the theory.
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This work was partially supported by NSF Grants DMS-1216740 and DMS-1620288.
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Karakashian, O., Collins, C. Two-Level Additive Schwarz Methods for Discontinuous Galerkin Approximations of the Biharmonic Equation. J Sci Comput 74, 573–604 (2018). https://doi.org/10.1007/s10915-017-0453-4
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DOI: https://doi.org/10.1007/s10915-017-0453-4