volume | PIER Journals

Abstract

A finite element method based on the first order system LL* (FOSLL*) approach is derived for time harmonic Maxwell's equations in three dimensional domains. The finite element solution is a potential for the original field in a sense that the original field U is given by U = L*u. The Maxwellian boundary data appears as natural boundary condition. Homogeneous Dirichlet boundary conditions for the potential must be imposed, and they are circumvented with weak enforcement of boundary conditions and it is proved that the sesquilinear form of the finite element system is elliptic in the space where the Dirichlet boundary conditions are satisfied weakly.

1. Monk, P., Finite Element Methods for Maxwell's Equations, Numerical Analysis and Scientific Computation Series, Clarendon Press, 2003.
doi:10.1093/acprof:oso/9780198508885.001.0001

2. Bochev, P., "Least-squares finite element methods for first-order elliptic systems," Int. J. Numer. Anal. Model., Vol. 1, No. 1, 49-64, 2004.

3. Jiang, B.-N., "The Least-squares Finite Element Method: Theory and Applications in Computational Fluid Dynamics and Electromagnetics," Scientific Computation, Springer, Berlin, New York, 1998.

4. Aziz, A. K., R. B. Kellogg, and A. B. Stephens, "Least squares methods for elliptic systems," Mathematics of Computation, Vol. 44, No. 169, 53-70, 1985.
doi:10.1090/S0025-5718-1985-0771030-5

5. Bochev, P. and M. Gunzburger, Least-squares Finite Element Methods, Applied Mathematical Sciences, Springer, 2009.
doi:10.1007/b13382

6. Bramble, J. H., T. V. Kolev, and J. E. Pasciak, "A least-squares approximation method for the time-harmonic Maxwell equations," Journal of Numerical Mathematics, Vol. 13, No. 4, 237-263, 2005.
doi:10.1515/156939505775248347

7. Cai, Z., T. A. Manteu®el, S. F. McCormick, and J. Ruge, "First-order system LL* (FOSLL*): Scalar elliptic partial differential equations," SIAM J. Numerical Analysis, Vol. 39, No. 4, 1418-1445, 2001.
doi:10.1137/S0036142900388049

8. Lee, E. and T. A. Manteuffel, "FOSLL* Method for the eddy current problem with three-dimensional edge singularities," SIAM J. Numerical Analysis, Vol. 45, No. 2, 787-809, 2007.
doi:10.1137/050647001

9. Petrovsky, I., Lectures on Partial Differential Equations, Cambridge University Press, 1954.

10. Picard, R., "On the low frequency asymptotics in electromagnetic theory," Journal Fur Die Reine Und Angewandte Mathematik, Vol. 1984, 50-73, 1984.

11. Picard, R., "On a structural observation in generalized electromagnetic theory," Journal of Mathematical Analysis and Applications, Vol. 110, No. 1, 247-264, 1985.
doi:10.1016/0022-247X(85)90348-8

12. Babu·ska, I., "The finite element method with Lagrangian multipliers," Numerische Mathematik, Vol. 20, No. 3, 179-192, 1973.
doi:10.1007/BF01436561

13. Balay, S., J. Brown, K. Buschelman, W. D. Gropp, D. Kaushik, M. G. Knepley, L. C. McInnes, B. F. Smith, and H. Zhang, "PETSc Web page,", 2013, http://www.mcs.anl.gov/petsc.

14. Costabel, M., "A coercive bilinear form for Maxwell's equation," J. Math. Anal. Appl., Vol. 157, No. 2, 527-541, 1991.
doi:10.1016/0022-247X(91)90104-8

15. Taskinen, M. and S. Vanska, "Current and charge integral equation formulations and Picard's extended Maxwell system," IEEE Transactions on Antennas and Propagation, Vol. 55, No. 12, 3495-3503, 2007.
doi:10.1109/TAP.2007.910363

16. Leis, R., Initial Boundary Value Problems in Mathematical Physics, Teubner, 1986.
doi:10.1007/978-3-663-10649-4

17. Girault, V. and P. Raviart, "Finite Element Methods for Navier-stokes Equations: Theory and Algorithms," Springer Series in Computational Mathematics, Springer-Verlag, 1986.

18. Braess, D., Finite Elements: Theory, Fast Solvers, and Applications in Solid Mechanics, Cambridge University Press, 2001.

19. Nedelec, J., "Mixed finite elements in R³," Numerische Mathematik, Vol. 35, No. 3, 315-341, 1980.
doi:10.1007/BF01396415

20. Logg, A., K.-A. Mardal, G. N. Wells, et al. Automated Solution of Differential Equations by the Finite Element Method, Springer, 2012.
doi:10.1007/978-3-642-23099-8

21. Picard, R., "An elementary proof for a compact imbedding result in generalized electromagnetic theory," Mathematische Zeitschrift, Vol. 187, 151-164, 1984.
doi:10.1007/BF01161700

22. Smith, K., "Inequalities for formally positive integro-differential forms," Bulletin of the American Mathematical Society, Vol. 67, 368-370, 1961.
doi:10.1090/S0002-9904-1961-10622-8

Mr. Juhani Kataja