# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
New block triangular preconditioner for linear systems arising from the discretized time-harmonic Maxwell equations. (English) Zbl 1197.78068
Summary: Based on the preconditioners presented by T. Rees and C. Greif [SIAM J. Sci. Comput. 29, No. 5, 1992–2007 (2007; Zbl 1155.65048)], we present a new block triangular preconditioner applied to the problem of solving linear systems arising from finite element discretization of the mixed formulation of the time-harmonic Maxwell equations $\left(k=0\right)$ in electromagnetic problems, since linear systems arising from the corresponding equations and methods have the same matrix block structure. Similar to spectral distribution of the preconditioners presented by Rees and Greif, this paper analyzes the corresponding spectral distribution of the new preconditioners considered in this paper. From the views of theories and applications, the presented preconditioners are as efficient as the preconditioners presented by Rees and Greif to apply. Moreover, numerical experiments are also reported to illustrate the efficiency of the presented preconditioners.

##### MSC:
 78M99 Basic mathematical methods in optics 65F08 Preconditioners for iterative methods
##### References:
 [1] Bergamaschi, L.; Gondzio, J.; Zilli, G.: Preconditioning indefinite systems in interior point methods for optimization, Computational optimization and applications 28, 149-171 (2004) · Zbl 1056.90137 · doi:10.1023/B:COAP.0000026882.34332.1b [2] Chen, Z.; Du, Q.; Zou, J.: Finite element methods with matching and nonmatching meshes for Maxwell equations with discontinuous coefficients, SIAM journal on numerical analysis 37, 1542-1570 (2000) · Zbl 0964.78017 · doi:10.1137/S0036142998349977 [3] Demkowicz, L.; Vardapetyan, L.: Modeling of electromagnetic absorption/scattering problems using hp-adaptive finite elements, Computer methods in applied mechanics and engineering 152, 103-124 (1998) · Zbl 0994.78011 · doi:10.1016/S0045-7825(97)00184-9 [4] Houston, P.; Schneebeli, A.; Schötzau, D.: Mixed discontinuous Galerkin approximation of the Maxwell operator: the definite case, Modélisation mathematique et analyse numérique 39, 727-754 (2005) · Zbl 1087.65106 · doi:10.1051/m2an:2005032 · doi:numdam:M2AN_2005__39_4_727_0 [5] Perugia, I.; Schötzau, D.; Monk, P.: Stabilized interior penalty methods for the time-harmonic Maxwell equations, Computer methods in applied mechanics and engineering 191, 4675-4697 (2002) · Zbl 1040.78011 · doi:10.1016/S0045-7825(02)00399-7 [6] Greif, C.; Schötzau, D.: Preconditioners for the discretized time-harmonic Maxwell equations in mixed form, Numerical linear algebra and its applications 14, 281-297 (2007) · Zbl 1199.78010 · doi:10.1002/nla.515 [7] Keller, C.; Gould, N. I. M.; Wathen, A. J.: Constraint preconditioning for indefinite linear systems, SIAM J. Matrix analysis and application 21, 1300-1317 (2000) · Zbl 0960.65052 · doi:10.1137/S0895479899351805 [8] Rees, T.; Greif, C.: A preconditioner for linear systems arising from interior point optimization methods, SIAM J. Scientific computing 29, 1992-2007 (2007) · Zbl 1155.65048 · doi:10.1137/060661673 [9] Van Der Vorst, H. A.: Iterative Krylov methods for large linear systems, Cambridge monographs on applied and computational mathematics (2003)