Solution of a~regularized problem for a~stationary magnetic field in a~non-homogeneous conducting medium by a~finite element method

I. A. Kremer ; M. V. Urev

Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 13 (2010) no. 1, pp. 33-49.

Résumé

In this paper, we substantiate the use of a vector finite element method for solving a regularized stationary magnetic problem, which is formulated in terms of a vector magnetic potential. To approximate the generalized solution, we make use of the Nedelec second kind vector elements of first order on tetrahedrons. Existence and uniqueness of the solution to a discrete regularized problem and its convergence to a generalized solution for the case of an inhomogeneous domain (according to electromagnetic properties) are justified. Some issues of the numerical solution to a discrete regularized problem are discussed. Approaches to optimize the algorithms are shown on a series of numerical experiments.

Bibliographie

[1] Ivanov M. I., Kateshov V. A., Kremer I. A., Urev M. V., “Reshenie trekhmernykh statsionarnykh zadach impulsnoi elektrorazvedki”, Avtometriya, 43:2 (2007), 22–32

[2] Ivanov M. I., Kateshov V. A., Kremer I. A., Urev M. V., “Reshenie trekhmernykh nestatsionarnykh zadach impulsnoi elektrorazvedki”, Avtometriya, 43:2 (2007), 33–44

[3] Kremer I. A., Urev M. V., “Metod regulyarizatsii statsionarnoi sistemy Maksvella v neodnorodnoi provodyaschei srede”, Sib. zhurn. vychisl. matematiki, 12:2 (2009), 161–170

[4] Girault V., Raviart P.-A., Finite Element Methods for Navier–Stokes Equations, Springer-Verlag, Berlin, 1986 | MR | Zbl

[5] Brezzi F., Fortin M., Mixed and Hybrid Finite Element Methods, Springer-Verlag, New York, 1991 | MR | Zbl

[6] Greif C., Schötzau D., “Preconditioners for the discretized time-harmonic Maxwell equations in mixed form”, Numer. Linear Algebra Appl., 14:4 (2007), 281–297 | DOI | MR | Zbl

[7] Chizhonkov E. V., Relaksatsionnye metody resheniya sedlovykh zadach, IVM RAN, M., 2002

[8] Hu Q., Zou J., “Substructuring preconditioners for saddle-point problems arising from Maxwell's equations in three dimensions”, Math. Comp., 73 (2003), 35–61 | DOI | MR

[9] Haase G., Kuhn M., Langer U., “Parallel multigrid 3D Maxwell solvers”, Parallel Computing, 27:6 (2001), 761–775 | DOI | MR | Zbl

[10] Dyuvo G., Lions Zh.-L., Neravenstva v mekhanike i fizike, Nauka, M., 1980 | MR

[11] Lions Zh.-L., Madzhenes E., Neodnorodnye granichnye zadachi i ikh prilozheniya, Mir, M., 1971 | Zbl

[12] Kopachevsii N. D., Krein S. G., Ngo Zui Kan, Operatornye metody v lineinoi gidrodinamike, Nauka, M., 1989 | MR

[13] Nedelec J. C., “A new family of mixed finite elements in $\mathbb R^3$”, Numer. Math., 50 (1986), 57–81 | DOI | MR | Zbl

[14] Syarle F., Metod konechnykh elementov dlya ellipticheskikh zadach, Mir, M., 1980 | MR

[15] Bermúdez A., Rodríguez R., Salgado P., “A finite element method with Lagrange multipliers for low-frequency harmonic Maxwell equations”, Math. Comp., 74 (2005), 123–151 | DOI | MR | Zbl

[16] Babuška I., “The finite element method for elliptic equations with discontinuous coefficients”, Computing, 5 (1970), 207–213 | DOI | MR | Zbl

[17] Alonso A., Valli A., “An optimal domain decomposition preconditioner for low-frequency time-harmonic Maxwell equations”, Math. Comp., 68 (1999), 607–631 | DOI | MR | Zbl

[18] Ciarlet P. (Jr.), Zou J., “Fully discrette finite element approaches for time-dependent Maxwell's equations”, Numer. Math., 82 (1999), 193–219 | DOI | MR | Zbl

[19] Chen Z., Du Q., Zou J., “Finite element methods with matching and non-matching meshes for Maxwell equations with discontinuous coefficients”, SIAM J. Numer. Anal., 37 (2000), 1542–1570 | DOI | MR | Zbl