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Analysis of a finite element PML approximation for the three dimensional time-harmonic Maxwell problem
Author(s):
James
H.
Bramble;
Joseph
E.
Pasciak.
Journal:
Math. Comp.
77
(2008),
1-10.
MSC (2000):
Primary 78M10, 65F10, 65N30
Posted:
September 18, 2007
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Abstract:
In our paper [Math. Comp. 76, 2007, 597-614] we considered the acoustic and electromagnetic scattering problems in three spatial dimensions. In particular, we studied a perfectly matched layer (PML) approximation to an electromagnetic scattering problem. We demonstrated both the solvability of the continuous PML approximations and the exponential convergence of the resulting solution to the solution of the original acoustic or electromagnetic problem as the layer increased. In this paper, we consider finite element approximation of the truncated PML electromagnetic scattering problem. Specifically, we consider approximations which result from the use of Nédélec (edge) finite elements. We show that the resulting finite element problem is stable and gives rise to quasi-optimal convergence when the mesh size is sufficiently small.
References:
-
- 1.
- C. Amrouche, C. Bernardi, M. Dauge, and V. Girault.
Vector potentials in three-dimensional non-smooth domains.
Math. Methods Appl. Sci., 21(9):823-864, 1998. MR 1626990 (99e:35037) - 2.
- G. Bao and H. Wu.
Convergence analysis of the perfectly matched layer problems for time-harmonic Maxwell's equations.
SIAM J. Numer. Anal., 43(5):2121-2143 (electronic), 2005. MR 2192334 - 3.
- J. H. Bramble and J. E. Pasciak.
Analysis of a finite PML approximation for the three dimensional time-harmonic maxwell and acoustic scattering problems. Math. Comp., 76 (2007), 597-614. - 4.
- F. Collino and P. Monk.
The perfectly matched layer in curvilinear coordinates.
SIAM J. Sci. Comp., 19(6):2061-2090, 1998. MR 1638033 (99e:78029) - 5.
- V. Girault and P. Raviart.
Finite Element Approximation of the Navier-Stokes Equations.
Lecture Notes in Math. 749, Springer-Verlag, New York, 1979. MR 0548867 (83b:65122). - 6.
- J. Gopalakrishnan and J. E. Pasciak.
Overlapping Schwarz preconditioners for indefinite time harmonic Maxwell equations.
Math. Comp., 72(241):1-15, 2003. MR 1933811 (2003i:78020) - 7.
- F. Kukuchi.
On a discrete compactness property for the Nédélec finite elements.
J. Fac. Sci. Univ. Tokyo, Sect. 1A, Math, 36:479-490, 1989. MR 1039483 (91h:65173) - 8.
- M. Lassas and E. Somersalo.
On the existence and convergence of the solution of PML equations.
Computing, 60(3):229-241, 1998. MR 1621305 (99a:65133) - 9.
- P. Monk.
A simple proof of convergence for an edge element discretization of maxwell's equations.
Lecture Notes in Comp. Sci. Engl., 28, Springer, Berlin, 2003. MR 1986135 (2004i:78024) - 10.
- P. Monk.
Finite element methods for Maxwell's equations.
Oxford Science Pub., Oxford, 2003. - 11.
- P. Monk and L. Demkowicz.
Discrete compactness and the approximation of Maxwell's equations in  .
Math. Comp., 70(234):507-523, 2001. MR 1709155 (2001g:65156) - 12.
- J. C. Nédélec.
Mixed finite elements in  .
Numer. Math., 35:315-341, 1980. MR 592160 (81k:65125) - 13.
- J. C. Nédélec.
A new family of mixed finite elements in  .
Numer. Math., 50:57-81, 1986. MR 864305 (88e:65145) - 14.
- A. Schatz.
An observation concerning Ritz-Galerkin methods with indefinite bilinear forms.
Math. Comp., 28:959-962, 1974. MR 0373326 (51:9526)
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Additional Information:
James
H.
Bramble
Affiliation:
Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368.
Email:
bramble@math.tamu.edu
Joseph
E.
Pasciak
Affiliation:
Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368.
Email:
pasciak@math.tamu.edu
DOI:
10.1090/S0025-5718-07-02037-6
PII:
S 0025-5718(07)02037-6
Keywords:
Maxwell's equations,
Helmholtz equation,
time-harmonic acoustic and electromagnetic scattering,
div-curl systems,
PML layer
Received by editor(s):
September 11, 2006
Received by editor(s) in revised form:
January 24, 2007
Posted:
September 18, 2007
Additional Notes:
This work was supported in part by the National Science Foundation through grant No. 0311902.
Copyright of article:
Copyright
2007,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
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