Domain decomposition algorithms for time-harmonic Maxwell equations with damping
ESAIM: Modélisation mathématique et analyse numérique, Tome 35 (2001) no. 4, pp. 825-848.

Three non-overlapping domain decomposition methods are proposed for the numerical approximation of time-harmonic Maxwell equations with damping (i.e., in a conductor). For each method convergence is proved and, for the discrete problem, the rate of convergence of the iterative algorithm is shown to be independent of the number of degrees of freedom.

Classification : 65N55, 65N30
Mots-clés : time-harmonic Maxwell equations, domain decomposition methods, edge finite elements 
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     title = {Domain decomposition algorithms for time-harmonic {Maxwell} equations with damping},
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Rodriguez, Ana Alonso; Valli, Alberto. Domain decomposition algorithms for time-harmonic Maxwell equations with damping. ESAIM: Modélisation mathématique et analyse numérique, Tome 35 (2001) no. 4, pp. 825-848. http://www.numdam.org/item/M2AN_2001__35_4_825_0/

[1] V.I. Agoshkov and V.I. Lebedev, Poincaré-Steklov operators and the methods of partition of the domain in variational problems, in Vychisl. Protsessy Sist. (Computational processes and systems), G.I. Marchuk, Ed., Nauka, Moscow 2 (1985) 173-227 (in Russian). | Zbl

[2] A. Alonso and A. Valli, Some remarks on the characterization of the space of tangential traces of H(rot;Ω) and the construction of an extension operator. Manuscripta Math. 89 (1996) 159-178. | Zbl

[3] A. Alonso and A. Valli, An optimal domain decomposition preconditioner for low-frequency time-harmonic Maxwell equations. Math. Comp. 68 (1999) 607-631. | Zbl

[4] A. Alonso and A. Valli, A domain decomposition approach for heterogeneous time-harmonic Maxwell equations. Comput. Methods Appl. Mech. Engrg. 143 (1997) 97-112. | Zbl

[5] A. Alonso, R.L. Trotta and A. Valli, Coercive domain decomposition algorithms for advection-diffusion equations and systems. J. Comput. Appl. Math. 96 (1998) 51-76. | Zbl

[6] L.C. Berselli, Some topics in fluid mechanics. Ph.D. thesis, Dipartimento di Matematica, Università di Pisa, Italy (1999).

[7] L.C. Berselli and F. Saleri, New substructuring domain decomposition methods for advection-diffusion equations. J. Comput. Appl. Math. 116 (2000) 201-220. | Zbl

[8] P.E. Bjørstad and O.B. Widlund, Iterative methods for the solution of elliptic problems on regions partitioned into substructures. SIAM J. Numer. Anal. 23 (1986) 1097-1120. | Zbl

[9] A. Bossavit, Électromagnétisme, en vue de la modélisation. Springer-Verlag, Paris (1993). | Zbl

[10] J.-F. Bourgat, R. Glowinski, P. Le Tallec and M. Vidrascu, Variational formulation and algorithm for trace operator in domain decomposition calculations, in Domain Decomposition Methods, T.F. Chan et al., Eds., SIAM, Philadelphia (1989) 3-16. | Zbl

[11] J.H. Bramble, J.E. Pasciak and A.H. Schatz, An iterative method for elliptic problems on regions partitioned into substructures. Math. Comp. 46 (1986) 361-369. | Zbl

[12] A. Buffa and P. Ciarlet, Jr., On traces for functional spaces related to Maxwell's equations Part I: An integration by parts formula in Lipschitz polyhedra. Math. Methods Appl. Sci. 24 (2001) 9-30. | Zbl

[13] A. Buffa and P. Ciarlet, Jr., On traces for functional spaces related to Maxwell's equations Part II: Hodge decompositions on the boundary of Lipschitz polyhedra and applications. Math. Meth. Appl. Sci. 24 (2001) 31-48. | Zbl

[14] M. Cessenat, Mathematical methods in electromagnetism: Linear theory and applications. World Scientific Pub. Co., Singapore (1996). | MR | Zbl

[15] P. Collino, G. Delbue, P. Joly and A. Piacentini, A new interface condition in the non-overlapping domain decomposition method for the Maxwell equation. Comput. Methods Appl. Mech. Engrg. 148 (1997) 195-207. | Zbl

[16] B. Després, P. Joly and J.E. Roberts, A domain decomposition method for the harmonic Maxwell equation, in Iterative Methods in Linear Algebra, R. Beaurvens and P. de Groen, Eds., North Holland, Amsterdam (1992) 475-484. | Zbl

[17] S. Kim, Domain decomposition iterative procedures for solving scalar waves in the frequency domain. Numer. Math. 79 (1998) 231-259. | Zbl

[18] R. Leis, Exterior boundary-value problems in mathematical physics, in Trends in Applications of Pure Mathematics to Mechanics 11, H. Zorski, Ed., Pitman, London (1979) 187-203. | Zbl

[19] L.D. Marini and A. Quarteroni, A relaxation procedure for domain decomposition methods using finite elements. Numer. Math. 55 (1989) 575-598. | Zbl

[20] P. Monk, A finite element method for approximating the time-harmonic Maxwell equations. Numer. Math. 63 (1992) 243-261. | Zbl

[21] J.C. Nédélec, Mixed finite elements in ℝ 3 . Numer. Math. 35 (1980) 315-341. | Zbl

[22] J.C. Nédélec, A new family of mixed finite elements in ℝ 3 . Numer. Math. 50 (1986) 57-81. | Zbl

[23] A. Quarteroni and A. Valli, Domain decomposition methods for partial differential equations. Oxford University Press, Oxford (1999). | MR | Zbl

[24] J.E. Santos, Global and domain-decomposed mixed methods for the solution of Maxwell's equations with application to magnetotellurics. Numer. Methods. Partial Differ. Equations 14 (1998) 407-437. | Zbl

[25] A. Toselli, Domain decomposition methods for vector field problems. Ph.D. thesis, Courant Institute, New York University, New York (1999).