Domain decomposition algorithms for time-harmonic Maxwell equations with damping
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 35 (2001) no. 4, p. 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
Keywords: time-harmonic Maxwell equations, domain decomposition methods, edge finite elements
@article{M2AN_2001__35_4_825_0,
     author = {Rodriguez, Ana Alonso and Valli, Alberto},
     title = {Domain decomposition algorithms for time-harmonic Maxwell equations with damping},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {35},
     number = {4},
     year = {2001},
     pages = {825-848},
     zbl = {0993.78018},
     mrnumber = {1863282},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2001__35_4_825_0}
}
Rodriguez, Ana Alonso; Valli, Alberto. Domain decomposition algorithms for time-harmonic Maxwell equations with damping. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 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 0596.35030

[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 0856.46019

[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 1043.78554

[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 0883.65096

[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 0935.65137

[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 0946.65126

[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 0615.65113

[9] A. Bossavit, Électromagnétisme, en vue de la modélisation. Springer-Verlag, Paris (1993). | Zbl 0787.65090

[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 0684.65094

[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 0595.65111

[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 0998.46012

[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 0976.46023

[14] M. Cessenat, Mathematical methods in electromagnetism: Linear theory and applications. World Scientific Pub. Co., Singapore (1996). | MR 1409140 | Zbl 0917.65099

[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 0902.65074

[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 0785.65117

[17] S. Kim, Domain decomposition iterative procedures for solving scalar waves in the frequency domain. Numer. Math. 79 (1998) 231-259. | Zbl 0926.65132

[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 0414.73082

[19] L.D. Marini and A. Quarteroni, A relaxation procedure for domain decomposition methods using finite elements. Numer. Math. 55 (1989) 575-598. | Zbl 0661.65111

[20] P. Monk, A finite element method for approximating the time-harmonic Maxwell equations. Numer. Math. 63 (1992) 243-261. | Zbl 0757.65126

[21] J.C. Nédélec, Mixed finite elements in 3 . Numer. Math. 35 (1980) 315-341. | Zbl 0419.65069

[22] J.C. Nédélec, A new family of mixed finite elements in 3 . Numer. Math. 50 (1986) 57-81. | Zbl 0625.65107

[23] A. Quarteroni and A. Valli, Domain decomposition methods for partial differential equations. Oxford University Press, Oxford (1999). | MR 1857663 | Zbl 0931.65118

[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 0918.65083

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