On vector finite element method for solution to electromagnetic problems

È. P. Shurina ; M. A. Gelber

Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 7 (2004) no. 1, pp. 79-95.

Résumé

The vector finite element method is a comparatively new approach, therefore for this method neither a general theory no computational scheme has been developed. The aim of the present paper is to analyze application of the method to solution of electromagnetic problems. Special vector variational formulations have been constructed depending on the problem. Interpolation properties of different types of elements are investigated both theoretically and experimentally.

Bibliographie

[1] Ambrosiano J. J., Brandon S. T., Löhner R., DeVore C. R., “Electromagnetics via the Taylor–Galerkin finite element method on unstructed grids”, J. Comput. Phys., 110 (1994), 310–319 | DOI | Zbl

[2] Assous F., Ciarlet P., Segre J., “Numerical solution to the time-dependent Maxwell equations in two-dimensional singular domains: the singular complement method”, J. Comput. Phys., 161 (2000), 218–249 | DOI | MR | Zbl

[3] Bom D., Discrete compactness and Fortin operator for edge elements, Report No AM187, Penn State Univercity, USA, 1999

[4] Bom D., Gastaldi L., Edge finite elements for the approximation of Maxwell resolvent operator, Submitted to RAIRO M2AN, 2001

[5] Bossavit A., Computational electromagnetism. Variational formulations, complementary, edge elements, Academic Press, Casebound, 1997 | MR

[6] Braess D., Finite elements. Theory, fast solvers, and applications in solid mechanics, Cambridge University Press, Cambridge, 2001 | MR | Zbl

[7] Brenner S. C., Scott L. R., The mathematical theory of finite elements methods, Springer-Verlag, New York, 2002 | MR

[8] Brezzi F., Fortin M., Mixed and hybrid finite element methods, Springer-Verlag, New York, 1991 | MR | Zbl

[9] Cioni J.-P., Fezoui L., Issautier D., “High order upwind schemes for solving time-domain Maxwell equations”, La Recherche Aérospatiale, 5 (1994), 319–328 | MR

[10] Demcowicz L., Monk P., “Discrete compactness and the approximation of Maxwell's equations in $R^3$”, Math. Comp., 70 (2000), 507–523 | DOI

[11] Girault V., Raviart P.-A., Finite element methods for Navier–Stokes equations, Springer, New York, 1986 | MR | Zbl

[12] Hiptmair R., “Finite elements in computational electromagnetism”, Acta Numerica, 11 (2002), 237–339 | DOI | MR | Zbl

[13] Igarashi H., “On the property of the curl-curl matrix in finite element analysis with edge elements”, IEEE Trans. on Magn., 37:5 (2001), 3129–3132 | DOI

[14] Iwashita Y., PISCESII: 2.5D RF cavity code with high accuracy, ICAP'98, September 14–18, 1998, Monterey Conference Center, Monterey, California, USA

[15] Kanayama H., Large-scale magnetic field analysis, Annual report of ADVENTURE project ADV-99-1, 1999 | Zbl

[16] Munz C.-D., Schneider R., Voß U., “A finite-volume method for the Maxwell equations in the time domain”, SIAM J. Sci. Comput., 22:2 (1996), 449–475 | DOI | MR

[17] Nédélec J. C., “Mixed finite elements in $R^3$”, Numer. Math., 35 (1980), 315–341 | DOI | MR | Zbl

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

[19] Perugia I., “A mixed formulation for 3D magnetostatic problems: theoretical analysis and faceedge finite element approximation”, Numer. Math., 84 (1999), 305–326 | DOI | MR | Zbl

[20] Polstyanko S. V., Lee J.-F., “Two-level hierarchical FEM method for modeling passive microwave devices”, J. Comput. Phys., 140 (1998), 400–420 | DOI | Zbl

[21] Ren Z., Razek A., “Boundary edge elements and spanning tree technique in three-dimensional electromagnetic field computation”, Int. J. Numer. Methods in Eng., 36 (1993), 2877–2893 | DOI | MR

[22] Webb R., “Edge elements and what they can do for you”, IEEE Trans. on Magn., 29:2 (1993), 1460–1465 | DOI

[23] White D. A., “Numerical modeling of optical gradient traps using the vector finite element method”, J. Comput. Phys., 159 (2000), 13–37 | DOI | Zbl

[24] Whitney H., Geometric integration theory, Princeton University Press, Princeton, 1957 | MR | Zbl

[25] Wu J.-Y., Lee R., “The advantages of triangular and tetrahedral edge elements for electromagnetic modeling with the finite-element method”, IEEE Trans. Antennas and Propagation, 45 (1997), 1431–1437 | DOI

[26] Yee K., “Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media”, IEEE Trans. Antennas and Propagation, 16 (1966), 302–307

[27] Laevskii Yu. M., Metod konechnykh elementov (osnovy teorii, zadachi), Novosibirskii gos. universitet, Novosibirsk, 1999 | MR

[28] Streng G., Fiks Dzh., Teoriya metoda konechnykh elementov, Mir, M., 1977 | MR

[29] Shurina E. P., Voitovich T. V., “Analiz algoritmov metodov konechnykh elementov i konechnogo ob'ema na neortogonalnykh setkakh pri reshenii uravnenii Nave–Stoksa”, Vychislitelnye tekhnologii / RAN. Sib. otd-nie. — Novosibirsk, 2:4 (1997), 84–104 | MR