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An agglomeration-based massively parallel non-overlapping additive Schwarz preconditioner for high-order discontinuous Galerkin methods on polytopic grids


Authors: Paola F. Antonietti, Paul Houston, Giorgio Pennesi and Endre Süli
Journal: Math. Comp. 89 (2020), 2047-2083
MSC (2010): Primary 65M50, 65M55, 65Y05
DOI: https://doi.org/10.1090/mcom/3510
Published electronically: February 18, 2020
MathSciNet review: 4109560
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Abstract: In this article we design and analyze a class of two-level non-overlapping additive Schwarz preconditioners for the solution of the linear system of equations stemming from discontinuous Galerkin discretizations of second-order elliptic partial differential equations on polytopic meshes. The preconditioner is based on a coarse space and a non-overlapping partition of the computational domain where local solvers are applied in parallel. In particular, the coarse space can potentially be chosen to be non-embedded with respect to the finer space; indeed it can be obtained from the fine grid by employing agglomeration and edge coarsening techniques. We investigate the dependence of the condition number of the preconditioned system with respect to the diffusion coefficient and the discretization parameters, i.e., the mesh size and the polynomial degree of the fine and coarse spaces. Numerical examples are presented which confirm the theoretical bounds.


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Additional Information

Paola F. Antonietti
Affiliation: MOX-Laboratory for Modeling and Scientific Computing, Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy
Email: paola.antonietti@polimi.it

Paul Houston
Affiliation: School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD, United Kingdom
Email: Paul.Houston@nottingham.ac.uk

Giorgio Pennesi
Affiliation: MOX-Laboratory for Modeling and Scientific Computing, Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy
Email: giorgio.pennesi@polimi.it

Endre Süli
Affiliation: Mathematical Institute, University of Oxford, Andrew Wiles Building, Woodstock Road, Oxford, OX2 6GG, United Kingdom
Email: endre.suli@maths.ox.ac.uk

DOI: https://doi.org/10.1090/mcom/3510
Keywords: Domain decomposition, polytopic grids, discontinuous Galerkin methods.
Received by editor(s): March 26, 2019
Received by editor(s) in revised form: October 10, 2019, and November 12, 2019
Published electronically: February 18, 2020
Additional Notes: The first and third authors were partially funded by the SIR Project n. RBSI14VT0S funded by MIUR - Italian Ministry of Education, Universities and Research. The first and third authors also acknowledge the financial support given by GNCS-INdAM.
Article copyright: © Copyright 2020 American Mathematical Society