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On the spectrum of Schur complements of 2D elastic clusters joined by rigid edge modes and hybrid domain decomposition

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Abstract

The hybrid FETI-DP method proposed by Klawonn and Rheinbach uses a two-level decomposition of the domain into subdomains and clusters. Here we give bounds on the regular condition number of the clusters obtained by interconnecting the Schur complements of square elastic subdomains by the average rigid body modes of adjacent edges. Using the angles of subspaces and bounds on the spectrum of the subdomains’ Schur complements, we show that the conditioning of clusters comprising \(m\times m\) square subdomains increases proportionally to m. The estimate supports the scalability of the unpreconditioned hybrid FETI-DP method for both linear and contact problems. The numerical experiments confirm the efficiency of a coarse grid split between the primal and dual variables and indicate that hybrid FETI-DP with large clusters is a competitive tool for solving huge elasticity problems.

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Authors and Affiliations

  1. Department of Applied Mathematics, Faculty of Electrical Engineering and Computer Science, VŠB-Technical University of Ostrava, Ostrava, Czech Republic

    Zdeněk Dostál & Oldřich Vlach

  2. IT4Innovations, National Supercomputing Center, VŠB-Technical University of Ostrava, Ostrava, Czech Republic

    Zdeněk Dostál, Tomáš Brzobohatý, Oldřich Vlach & Lubomír Říha

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  1. Zdeněk Dostál
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  2. Tomáš Brzobohatý
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  3. Oldřich Vlach
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  4. Lubomír Říha
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Correspondence to Zdeněk Dostál.

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Dedicated to the memory of our dear colleague and friend Alexandros Markopoulos, whose early experiments drew our attention to these problems.

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This work was supported by the Ministry of Education, Youth and Sports of the Czech Republic through the e-INFRA CZ (ID:90140).

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Dostál, Z., Brzobohatý, T., Vlach, O. et al. On the spectrum of Schur complements of 2D elastic clusters joined by rigid edge modes and hybrid domain decomposition. Numer. Math. 152, 41–66 (2022). https://doi.org/10.1007/s00211-022-01307-x

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