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.
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Arnold, D.N., Falk, R.S., Winther, R.: Multigrid in H(div) and H(curl). Numer. Math. 85(2), 197–217 (2000)
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)
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)
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)
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)
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)
Dautray, R., Lions, J.L.: Mathematical analysis and numerical methods for science and technology, vol. 2. Springer, Berlin (1988)
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)
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)
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)
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)
Girault, V., Raviart, P.A.: Finite element methods for Navier-Stokes equations, Springer Series in computational mathematics, vol. 5. Springer, Berlin (1986). Theory and algorithms
Hiptmair, R.: Multigrid method for Maxwell’s equations. SIAM J. Numer. Anal. 36(1), 204–225 (1999)
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)
Hiptmair, R., Xu, J.: Nodal auxiliary space preconditioning in H(curl) and H(div) spaces. SIAM J. Numer. Anal. 45, 2483–2509 (2007)
John, F.: Rotation and strain. Comm. Pure Appl. Math. 14, 391–413 (1961)
Jones, P.W.: Quasiconformal mappings and extendability of functions in Sobolev spaces. Acta Math. 147(1-2), 71–88 (1981)
Karypis, G., Kumar, V.: A fast and high quality multilevel scheme for partitioning irregular graphs. SIAM J. Sci. Comput. 20(1), 359–392 (1998)
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)
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)
Leis, R.: Initial-boundary value problems in mathematical physics. B. G. Teubner. Wiley, Stuttgart (1986)
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)
Nédeléc, J.C.: Mixed finite elements in ℝ3. Numer. Math. 35, 315–341 (1980)
Saad, Y.: Iterative methods for sparse linear systems, 2nd edn. SIAM (2003)
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)
Toselli, A.: Overlapping Schwarz methods for Maxwell’s equations in three dimensions. Numer. Math. 86, 733–752 (2000)
Toselli, A., Widlund, O.B.: Domain decomposition methods-algorithms and theory, Springer Ser. Comput. Math., vol. 34. Springer (2005)
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)
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)
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)
The author would like to thank his former advisor, Prof. Olof Widlund, for his valuable guidance and suggestions during the work on this article. The author gratefully acknowledges the institutional support for the project B5A28 subscribed to the Vice-Rectory for Research, University of Costa Rica.
This work was supported in part by the National Science Foundation Grant Nos. DMS-1216564 and DMS-1522736.
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Calvo, J.G. A new coarse space for overlapping Schwarz algorithms for H(curl) problems in three dimensions with irregular subdomains. Numer Algor 83, 885–899 (2020). https://doi.org/10.1007/s11075-019-00707-9
- Domain decomposition
- Overlapping Schwarz algorithms
- Irregular subdomain boundaries
- Maxwell’s equations
- Discontinuous coefficients
Mathematics Subject Classification (2010)