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Numerical Algorithms
Published in: Numerical Algorithms 3/2020

26-04-2019 | Original Paper

A new coarse space for overlapping Schwarz algorithms for H(curl) problems in three dimensions with irregular subdomains

Author: Juan G. Calvo

Published in: Numerical Algorithms | Issue 3/2020

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Abstract

A new coarse space for a two-level overlapping Schwarz algorithm is presented for problems posed in three dimensions in the space H(curl, Ω). Previous studies for these methods are very restrictive about the geometry of the subdomains while this new space is well defined for general subdomains. The coarse space is based on energy minimization and its dimension equals the number of interior subdomain edges. Local direct solvers are used on the overlapping subdomains. The algorithm can be defined for any subdomain geometry and works for highly discontinuous coefficient distributions. Numerical experiments with irregular subdomains and different coefficient distributions are presented. The algorithm appears very promising even for random and discontinuous values of the coefficients.
previous article A Picard-type iterative algorithm for general variational inequalities and nonexpansive mappings
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Literature
1.
2.
Beck, R., Hiptmair, R., Hoppe, R.H.W., Wohlmuth, B.: Residual based a posteriori error estimators for eddy current computation. ESAIM: Math. Model. Numer. Anal. 34(1), 159–182 (2000)MathSciNetCrossRef
3.
Bossavit, A.: Discretization of electromagnetic problems: the “generalized finite differences” approach. In: Handbook of numerical analysis. Vol. XIII, Handb. Numer. Anal., XIII, pp 105–197. North-Holland, Amsterdam (2005)
4.
Calvo, J.G.: A two-level overlapping Schwarz method for H(curl) in two dimensions with irregular subdomains. Electron. Trans. Numer. Anal. 44, 497–521 (2015)MathSciNetMATH
5.
Calvo, J.G.: A BDDC algorithm with deluxe scaling for H(curl) in two dimensions with irregular subdomains. Math. Comp. 85(299), 1085–1111 (2016)MathSciNetCrossRef
6.
Calvo, J.G., Widlund, O.B.: An adaptive choice of primal constraints for BDDC domain decomposition algorithms. Electron. Trans. Numer. Anal. 45, 524–544 (2016)MathSciNetMATH
7.
Dautray, R., Lions, J.L.: Mathematical analysis and numerical methods for science and technology, vol. 2. Springer, Berlin (1988)CrossRef
8.
Dohrmann, C.R., Klawonn, A., Widlund, O.B.: Domain decomposition for less regular subdomains: Overlapping Schwarz in two dimensions. SIAM J. Numer. Anal. 46(5), 2153–2168 (2008)MathSciNetCrossRef
9.
Dohrmann, C.R., Widlund, O.B.: An alternative coarse space for irregular subdomains and an overlapping Schwarz algorithm for scalar elliptic problems in the plane. SIAM J. Numer. Anal. 50(5), 2522–2537 (2012)MathSciNetCrossRef
10.
Dohrmann, C.R., Widlund, O.B.: An iterative substructuring algorithm for two-dimensional problems in H(curl). SIAM J. Numer. Anal. 50(3), 1004–1028 (2012)MathSciNetCrossRef
11.
Dohrmann, C.R., Widlund, O.B.: A BDDC algorithm with deluxe scaling for three-dimensional H(curl) problems. Comm. Pure Appl. Math 69(4), 745–770 (2016)MathSciNetCrossRef
12.
Girault, V., Raviart, P.A.: Finite element methods for Navier-Stokes equations, Springer Series in computational mathematics, vol. 5. Springer, Berlin (1986). Theory and algorithmsCrossRef
13.
14.
Hiptmair, R., Toselli, A.: Overlapping and multilevel Schwarz methods for vector valued elliptic problems in three dimensions. In: Bjørstad, P., Luskin, M. (eds.) Parallel solution of Partial Differential Equations, vol. 120 of IMA Vol. Math. Appl., pp. 181–208. Springer (2000)
15.
Hiptmair, R., Xu, J.: Nodal auxiliary space preconditioning in H(curl) and H(div) spaces. SIAM J. Numer. Anal. 45, 2483–2509 (2007)MathSciNetCrossRef
16.
17.
Jones, P.W.: Quasiconformal mappings and extendability of functions in Sobolev spaces. Acta Math. 147(1-2), 71–88 (1981)MathSciNetCrossRef
18.
Karypis, G., Kumar, V.: A fast and high quality multilevel scheme for partitioning irregular graphs. SIAM J. Sci. Comput. 20(1), 359–392 (1998)MathSciNetCrossRef
19.
Klawonn, A., Rheinbach, O., Widlund, O.B.: An analysis of a FETI-DP algorithm on irregular subdomains in the plane. SIAM J. Numer. Anal. 46(5), 2484–2504 (2008)MathSciNetCrossRef
20.
Lashuk, I.V., Vassilevski, P.S.: The construction of the coarse de Rham complexes with improved approximation properties. Comput. Methods Appl. Math. 14(2), 257–303 (2014)MathSciNetCrossRef
21.
Leis, R.: Initial-boundary value problems in mathematical physics. B. G. Teubner. Wiley, Stuttgart (1986)CrossRef
22.
Müller, C.: Foundations of the mathematical theory of electromagnetic waves. Revised and enlarged translation from the German. Die Grundlehren der mathematischen Wissenschaften Band, vol. 155. Springer, New York-Heidelberg (1969)
23.
24.
Saad, Y.: Iterative methods for sparse linear systems, 2nd edn. SIAM (2003)
25.
Spillane, N., Dolean, V., Hauret, P., Nataf, F., Pechstein, C., Scheichl, R.: Abstract robust coarse spaces for systems of PDEs via generalized eigenproblems in the overlaps. Numer. Math. 126, 741–770 (2014)MathSciNetCrossRef
26.
Toselli, A.: Overlapping Schwarz methods for Maxwell’s equations in three dimensions. Numer. Math. 86, 733–752 (2000)MathSciNetCrossRef
27.
Toselli, A., Widlund, O.B.: Domain decomposition methods-algorithms and theory, Springer Ser. Comput. Math., vol. 34. Springer (2005)
28.
Toselli, A., Widlund, O.B., Wohlmuth, B.: An iterative substructuring method for Maxwell’s equations in two dimensions. Math. Comp. 70(235), 935–949 (2001)MathSciNetCrossRef
29.
Widlund, O.B.: Accommodating irregular subdomains in domain decomposition theory. In: Bercovier, M., Gander, M.J., Kornhuber, R., Widlund, O.B. (eds.) Domain Decomposition Methods in Science and Engineering XVIII, Lecture Notes in Computational Science and Engineering, vol. 70, pp. 87–98. Springer (2009)
30.
Zampini, S.: Adaptive BDDC deluxe methods for H(curl). In: Proceedings of the XXIII Domain Decomposition conference, Lecture Notes in Computational Science and Engineering, vol. 116, pp. 257–264. Springer (2016)
Metadata
Title
A new coarse space for overlapping Schwarz algorithms for H(curl) problems in three dimensions with irregular subdomains
Author
Juan G. Calvo
Publication date
26-04-2019
Publisher
Springer US
Published in
Numerical Algorithms / Issue 3/2020
Print ISSN: 1017-1398
Electronic ISSN: 1572-9265
DOI
https://doi.org/10.1007/s11075-019-00707-9

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