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Highly scalable hybrid domain decomposition method for the solution of huge scalar variational inequalities

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Abstract

The unpreconditioned hybrid domain decomposition method was recently shown to be a competitive solver for linear elliptic PDE problems discretized by structured grids. Here, we plug H-TFETI-DP (hybrid total finite element tearing and interconnecting dual primal) method into the solution of huge boundary elliptic variational inequalities. We decompose the domain into subdomains that are discretized and then interconnected partly by Lagrange multipliers and partly by edge averages. After eliminating the primal variables, we get a quadratic programming problem with a well-conditioned Hessian and bound and equality constraints that is effectively solvable by specialized algorithms. We prove that the procedure enjoys optimal, i.e., asymptotically linear complexity. The analysis uses recently established bounds on the spectrum of the Schur complements of the clusters interconnected by edge/face averages. The results extend the scope of scalability of massively parallel algorithms for the solution of variational inequalities and show the outstanding efficiency of the H-TFETI-DP coarse grid split between the primal and dual variables.

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Funding

This work was supported by The Ministry of Education, Youth, and Sports from the National Programme of Sustainability (NPS II) project “IT4Innovations excellence in science - LQ1602” and by the IT4Innovations infrastructure which is supported from the Large Infrastructures for Research, Experimental Development and Innovations project “e-INFRA CZ– LM2018140.” The second and third authors were supported by the grants 19-11441S of the Czech Science Foundation (GACR), SGS SP2021/103 of the VŠB – Technical University of Ostrava and “RRC/10/2019” Support for Science and Research in the Moravia–Silesia Region 2.

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Authors and Affiliations

  1. Department of Applied Mathematics and IT4Innovations-National Supercomputing Center, VSB-Technical University of Ostrava, 17. listopadu 15, 70833, Ostrava, Czech Republic

    Zdeněk Dostál & Oldřich Vlach

  2. Department of Applied Mathematics, VSB-TU Ostrava and Institue of Geonics ASCR, Ostrava, Czech Republic

    David Horák & Jakub Kružík

  3. IT4Innovations-National Supercomputing Center, VSB-TU Ostrava, Ostrava, Czech Republic

    Tomáš Brzobohatý

Authors
  1. Zdeněk Dostál
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  2. David Horák
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  3. Jakub Kružík
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  4. Tomáš Brzobohatý
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  5. Oldřich Vlach
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Correspondence to Zdeněk Dostál.

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Dostál, Z., Horák, D., Kružík, J. et al. Highly scalable hybrid domain decomposition method for the solution of huge scalar variational inequalities. Numer Algor 91, 773–801 (2022). https://doi.org/10.1007/s11075-022-01281-3

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  • DOI: https://doi.org/10.1007/s11075-022-01281-3

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