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Keywords:
time-harmonic Maxwell equations; non-homogeneous conductivities; three- dimensional problem; error estimation; finite element approximation; numerical experiments; solution theory
Summary:
The solvability of time-harmonic Maxwell equations in the 3D-case with nonhomogeneous conductivities is considered by adapting Sobolev space technique and variational formulation of the problem in question. Moreover, a finite element approximation is presented in the 3D-case together with an error estimate in the energy norm. Some remarks are given to the 2D-problem arising from geophysics.
References:
[1] V. Bezvoda K. Segeth: Mathematical modeling in electromagnetic prospecting methods. Charles Univ., Prague, 1982.
[2] E. B. Byhovskiy: Solution of a mixed problem for the system of Maxwell equations in case of ideally conductive boundary. Vestnik Leningrad. Univ. Mat. Mekh. Astronom. 12 (1957), 50-66. MR 0098567
[3] V. Červ K. Segeth: A comparison of the accuracy of the finite-difference solution to boundary-value problems for the Helmholtz equation obtained by direct and iterative methods. Apl. Mat. 27 (1982), 375-390. MR 0674982
[4] P. G. Ciarlet: The finite element method for elliptic problems. North-Holland, Amsterdam, New York, Oxford, 1978. MR 0520174 | Zbl 0383.65058
[5] D. Colton L. Päivärinta: Far field patterns and the inverse scattering problem for electromagnetic waves in an inhomogeneous medium. Math. Proc. Cambridge Philos. Soc. 103 (1988), 561-575. DOI 10.1017/S0305004100065154 | MR 0932679
[6] G. Duvaut J. L. Lions: Inequalities in mechanics and physics. Springer-Verlag, Berlin, 1976. MR 0521262
[7] V. Girault P. A. Raviart: Finite element methods for Navier-Stokes equations. Springer- Verlag, Berlin, Heidelberg, New York, Tokyo, 1986. MR 0851383
[8] P. Grisvard: Boundary value problems in nonsmooth domains. Lecture Notes 19, Univ. of Maryland, Dep. of Math., College Park, 1980.
[9] G. Heindl: Interpolation and approximation by piecewise quadratic $C^1$-functions of two variables. in Multivariate approximation theory, ed. by W. Schempp and K. Zeller. ISNM vol. 51, Birkhäuser, Basel, 1979, pp. 146-161. MR 0560670
[10] E. Hewitt K. Stromberg: Real and abstract analysis. Springer-Verlag, New York, Heidelberg, Berlin, 1975. MR 0367121
[11] J. Kadlec: On the regularity of the solution of the Poisson problem on a domain with boundary locally similar to the boundary of convex open set. Czechoslovak Math. J. 14 (1964), 386-393. MR 0170088
[12] F. Kikuchi: An isomorphic property of two Hubert spaces appearing in electromagnetism. Japan J. Appl. Math. 3 (1986), 53-58. DOI 10.1007/BF03167091 | MR 0899213
[13] M. Křížek P. Neittaanmäki: Solvability of a first order system in three-dimensional nonsmooth domains. Apl. Mat. 30 (1985), 307-315. MR 0795991
[14] S. Mareš, al.: Introduction to applied geophysics. (Czech). SNTL, Prague, 1979.
[15] J. Nečas: Les méthodes directes en théorie des équations elliptiques. Academia, Prague, 1967. MR 0227584
[16] P. Neittaanmäki R. Picard: Error estimates for the finite element approximation to a Maxwell-type boundary value problem. Numer. Funct. Anal. Optim. 2 (1980), 267-285. DOI 10.1080/01630568008816057 | MR 0588947
[17] J. Saranen: On an inequality of Friedrichs. Math. Scand. 51 (1982), 310-322. DOI 10.7146/math.scand.a-11983 | MR 0690534 | Zbl 0524.35100
[18] R. S. Varga: Matrix iterative analysis. Prentice-Hall, Englewood Cliffs, New Jersey, 1962. MR 0158502
[19] V. V. Voevodin, Ju. A. Kuzněcov: Matrices and computation. (Russian). Nauka, Moscow, 1984.
[20] Ch. Weber: A local compactness theorem for Maxwell's equations. Math. Methods Appl. Sci. 2 (1980), 12-25. DOI 10.1002/mma.1670020103 | MR 0561375 | Zbl 0432.35032
[21] Ch. Weber: Regularity theorems for Maxwell's equations. Math. Methods Appl. Sci. 3 (1981), 523-536. DOI 10.1002/mma.1670030137 | MR 0657071 | Zbl 0477.35020
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