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Uniform Subspace Correction Preconditioners for Discontinuous Galerkin Methods with hp-Refinement

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Abstract

In this paper, we develop subspace correction preconditioners for discontinuous Galerkin (DG) discretizations of elliptic problems with hp-refinement. These preconditioners are based on the decomposition of the DG finite element space into a conforming subspace, and a set of small nonconforming edge spaces. The conforming subspace is preconditioned using a matrix-free low-order refined technique, which in this work, we extend to the hp-refinement context using a variational restriction approach. The condition number of the resulting linear system is independent of the granularity of the mesh h, and the degree of the polynomial approximation p. The method is amenable to use with meshes of any degree of irregularity and arbitrary distribution of polynomial degrees. Numerical examples are shown on several test cases involving adaptively and randomly refined meshes, using both the symmetric interior penalty method and the second method of Bassi and Rebay (BR2).

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Acknowledgements

This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344 and was partially supported by the LLNL-LDRD Program under Project No. 20-ERD-002 (LLNL-JRNL-814157). This document was prepared as an account of work sponsored by an agency of the United States government. Neither the United States government nor Lawrence Livermore National Security, LLC, nor any of their employees makes any warranty, expressed or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States government or Lawrence Livermore National Security, LLC. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States government or Lawrence Livermore National Security, LLC, and shall not be used for advertising or product endorsement purposes.

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    Will Pazner & Tzanio Kolev

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  1. Will Pazner
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Pazner, W., Kolev, T. Uniform Subspace Correction Preconditioners for Discontinuous Galerkin Methods with hp-Refinement. Commun. Appl. Math. Comput. 4, 697–727 (2022). https://doi.org/10.1007/s42967-021-00136-3

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