A locally divergence-free nonconforming finite element method for the time-harmonic Maxwell equations

Brenner, Susanne; Li, Fengyan; Sung, Li-yeng

doi:10.1090/S0025-5718-06-01950-8

A locally divergence-free nonconforming finite element method for the time-harmonic Maxwell equations
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by Susanne C. Brenner, Fengyan Li and Li-yeng Sung PDF

Math. Comp. 76 (2007), 573-595 Request permission

Abstract:

A new numerical method for computing the divergence-free part of the solution of the time-harmonic Maxwell equations is studied in this paper. It is based on a discretization that uses the locally divergence-free Crouzeix-Raviart nonconforming $P_1$ vector fields and includes a consistency term involving the jumps of the vector fields across element boundaries. Optimal convergence rates (up to an arbitrary positive $\epsilon$) in both the energy norm and the $L_2$ norm are established on graded meshes. The theoretical results are confirmed by numerical experiments.

References

Thomas Apel, Anisotropic finite elements: local estimates and applications, Advances in Numerical Mathematics, B. G. Teubner, Stuttgart, 1999. MR 1716824
F. Assous, P. Ciarlet Jr., and E. Sonnendrücker, Resolution of the Maxwell equations in a domain with reentrant corners, RAIRO Modél. Math. Anal. Numér. 32 (1998), no. 3, 359–389 (English, with English and French summaries). MR 1627135, DOI 10.1051/m2an/1998320303591
J. H. Bramble and S. R. Hilbert, Estimation of linear functionals on Sobolev spaces with application to Fourier transforms and spline interpolation, SIAM J. Numer. Anal. 7 (1970), 112–124. MR 263214, DOI 10.1137/0707006
S.C. Brenner and C. Carstensen. Finite Element Methods. In E. Stein, R. de Borst, and T.J.R. Hughes, editors, Encyclopedia of Computational Mechanics, pages 73–118. Wiley, Weinheim, 2004.
Susanne C. Brenner and L. Ridgway Scott, The mathematical theory of finite element methods, 2nd ed., Texts in Applied Mathematics, vol. 15, Springer-Verlag, New York, 2002. MR 1894376, DOI 10.1007/978-1-4757-3658-8
Philippe G. Ciarlet, The finite element method for elliptic problems, Studies in Mathematics and its Applications, Vol. 4, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. MR 0520174
Martin Costabel and Monique Dauge, Singularities of electromagnetic fields in polyhedral domains, Arch. Ration. Mech. Anal. 151 (2000), no. 3, 221–276. MR 1753704, DOI 10.1007/s002050050197
Martin Costabel and Monique Dauge, Weighted regularization of Maxwell equations in polyhedral domains. A rehabilitation of nodal finite elements, Numer. Math. 93 (2002), no. 2, 239–277. MR 1941397, DOI 10.1007/s002110100388
M. Crouzeix and P.-A. Raviart, Conforming and nonconforming finite element methods for solving the stationary Stokes equations. I, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge 7 (1973), no. R-3, 33–75. MR 343661
C. Cuvelier, A. Segal, and A. A. van Steenhoven, Finite element methods and Navier-Stokes equations, Mathematics and its Applications, vol. 22, D. Reidel Publishing Co., Dordrecht, 1986. MR 850259, DOI 10.1007/978-94-010-9333-0
Monique Dauge, Elliptic boundary value problems on corner domains, Lecture Notes in Mathematics, vol. 1341, Springer-Verlag, Berlin, 1988. Smoothness and asymptotics of solutions. MR 961439, DOI 10.1007/BFb0086682
Vivette Girault and Pierre-Arnaud Raviart, Finite element methods for Navier-Stokes equations, Springer Series in Computational Mathematics, vol. 5, Springer-Verlag, Berlin, 1986. Theory and algorithms. MR 851383, DOI 10.1007/978-3-642-61623-5
P. Grisvard, Elliptic problems in nonsmooth domains, Monographs and Studies in Mathematics, vol. 24, Pitman (Advanced Publishing Program), Boston, MA, 1985. MR 775683
Paul Houston, Ilaria Perugia, Anna Schneebeli, and Dominik Schötzau, Interior penalty method for the indefinite time-harmonic Maxwell equations, Numer. Math. 100 (2005), no. 3, 485–518. MR 2194528, DOI 10.1007/s00211-005-0604-7
Peter Monk, Finite element methods for Maxwell’s equations, Numerical Mathematics and Scientific Computation, Oxford University Press, New York, 2003. MR 2059447, DOI 10.1093/acprof:oso/9780198508885.001.0001
I. Perugia, D. Schötzau, and P. Monk, Stabilized interior penalty methods for the time-harmonic Maxwell equations, Comput. Methods Appl. Mech. Engrg. 191 (2002), no. 41-42, 4675–4697. MR 1929626, DOI 10.1016/S0045-7825(02)00399-7
Alfred H. Schatz, An observation concerning Ritz-Galerkin methods with indefinite bilinear forms, Math. Comp. 28 (1974), 959–962. MR 373326, DOI 10.1090/S0025-5718-1974-0373326-0
François Thomasset, Implementation of finite element methods for Navier-Stokes equations, Springer Series in Computational Physics, Springer-Verlag, New York-Berlin, 1981. MR 720192