Mathematics of Computation

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

Author(s): Susanne C. Brenner; Fengyan Li; Li-yeng Sung.
Journal: Math. Comp. 76 (2007), 573-595.
MSC (2000): Primary 65N30, 65N15, 35Q60
Posted: December 27, 2006
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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

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

norm are established on graded meshes. The theoretical results are confirmed by numerical experiments.

1.: Th. Apel.
Anisotropic Finite Elements.
Teubner, Stuttgart, 1999. MR 1716824 (2000k:65002)
2.: F. Assous, P. Ciarlet, Jr., and E. Sonnendrücker.
Resolution of the Maxwell equation in a domain with reentrant corners.
MAN, 32:359-389, 1998. MR 1627135 (2000b:78004)
3.: J.H. Bramble and S.R. Hilbert.
Estimation of linear functionals on Sobolev spaces with applications to Fourier transforms and spline interpolation.
SIAM J. Numer. Anal., 7:113-124, 1970. MR 0263214 (41:7819)
4.: 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.
5.: S.C. Brenner and L.R. Scott.
The Mathematical Theory of Finite Element Methods Second Edition.
Springer-Verlag, New York-Berlin-Heidelberg, 2002. MR 1894376 (2003a:65103)
6.: P.G. Ciarlet.
The Finite Element Method for Elliptic Problems.
North-Holland, Amsterdam, 1978. MR 0520174 (58:25001)
7.: M. Costabel and M. Dauge.
Singularities of electromagnetic fields in polyhedral domains.
Arch. Ration. Mech. Anal., 151:221-276, 2000. MR 1753704 (2002c:78005)
8.: M. Costabel and M. Dauge.
Weighted regularization of Maxwell equations in polyhedral domains.
Numer. Math., 93:239-277, 2002. MR 1941397 (2004a:78039)
9.: M. Crouzeix and P.-A. Raviart.
Conforming and nonconforming finite element methods for solving the stationary Stokes equations I.
RAIRO Anal. Numér., 7:33-75, 1973. MR 0343661 (49:8401)
10.: C. Cuvelier, A. Segal, and A.A. van Steenhoven.
Finite Element Methods and Navier-Stokes Equations.
D. Reidel Publishing Company, Dordrecht, 1986. MR 0850259 (88g:65106)
11.: M. Dauge.
Elliptic Boundary Value Problems on Corner Domains, Lecture Notes in Mathematics 1341.
Springer-Verlag, Berlin-Heidelberg, 1988. MR 0961439 (91a:35078)
12.: V. Girault and P.-A. Raviart.
Finite Element Methods for Navier-Stokes Equations. Theory and Algorithms.
Springer-Verlag, Berlin, 1986. MR 0851383 (88b:65129)
13.: P. Grisvard.
Elliptic Problems in Nonsmooth Domains.
Pitman, Boston, 1985. MR 0775683 (86m:35044)
14.: P. Houston, I. Perugia, A. Schneebeli, and D. Schötzau.
Interior penalty method for the indefinite time-harmonic Maxwell equations.
Numer. Math., 100:485-518, 2005. MR 2194528 (2006k:65323)
15.: P. Monk.
Finite Element Methods for Maxwell's Equations.
Oxford University Press, New York, 2003. MR 2059447 (2005d:65003)
16.: I. Perugia, D. Schötzau, and P. Monk.
Stabilized interior penalty methods for the time-harmonic Maxwell equations.
Comput. Methods. Appl. Mech. Engrg., 191:4675-4697, 2002. MR 1929626 (2003j:78058)
17.: A. Schatz.
An observation concerning Ritz-Galerkin methods with indefinite bilinear forms.
Math. Comp., 28:959-962, 1974. MR 0373326 (51:9526)
18.: F. Thomasset.
Implementation of Finite Element Methods for Navier-Stokes Equations.
Springer-Verlag, New York-Berlin, 1981. MR 0720192 (84k:76015)

Retrieve articles in Mathematics of Computation with MSC (2000): 65N30, 65N15, 35Q60

Susanne C. Brenner
Affiliation: Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208
Address at time of publication: Department of Mathematics and Center for Computation and Technology, Louisiana State University, Baton Rouge, Louisiana 70803
Email: brenner@math.lsu.edu

Fengyan Li
Affiliation: Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208
Address at time of publication: Department of Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, New York 12180
Email: lif@rpi.edu

Li-yeng Sung
Affiliation: Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208
Address at time of publication: Department of Mathematics, Louisiana State University, Baton Rouge, Louisiana 70803
Email: sung@math.lsu.edu

DOI: 10.1090/S0025-5718-06-01950-8
PII: S 0025-5718(06)01950-8
Keywords: Time-harmonic Maxwell equations, nonconforming finite element methods
Received by editor(s): August 8, 2005
Received by editor(s) in revised form: April 19, 2006
Posted: December 27, 2006
Additional Notes: The work of the first author was supported in part by the National Science Foundation under Grant No. DMS-03-11790.
Copyright of article: Copyright 2006, American Mathematical Society

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