Analysis of a finite element PML approximation for the three dimensional time-harmonic Maxwell problem
HTML articles powered by AMS MathViewer
- by James H. Bramble and Joseph E. Pasciak PDF
- Math. Comp. 77 (2008), 1-10 Request permission
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
- C. Amrouche, C. Bernardi, M. Dauge, and V. Girault, Vector potentials in three-dimensional non-smooth domains, Math. Methods Appl. Sci. 21 (1998), no. 9, 823–864 (English, with English and French summaries). MR 1626990, DOI 10.1002/(SICI)1099-1476(199806)21:9<823::AID-MMA976>3.0.CO;2-B
- Gang Bao and Haijun Wu, Convergence analysis of the perfectly matched layer problems for time-harmonic Maxwell’s equations, SIAM J. Numer. Anal. 43 (2005), no. 5, 2121–2143. MR 2192334, DOI 10.1137/040604315
- 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.
- Francis Collino and Peter Monk, The perfectly matched layer in curvilinear coordinates, SIAM J. Sci. Comput. 19 (1998), no. 6, 2061–2090. MR 1638033, DOI 10.1137/S1064827596301406
- V. Girault and P.-A. Raviart, Finite element approximation of the Navier-Stokes equations, Lecture Notes in Mathematics, vol. 749, Springer-Verlag, Berlin-New York, 1979. MR 548867
- Jayadeep Gopalakrishnan and Joseph E. Pasciak, Overlapping Schwarz preconditioners for indefinite time harmonic Maxwell equations, Math. Comp. 72 (2003), no. 241, 1–15. MR 1933811, DOI 10.1090/S0025-5718-01-01406-5
- Fumio Kikuchi, On a discrete compactness property for the Nédélec finite elements, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 36 (1989), no. 3, 479–490. MR 1039483
- M. Lassas and E. Somersalo, On the existence and convergence of the solution of PML equations, Computing 60 (1998), no. 3, 229–241. MR 1621305, DOI 10.1007/BF02684334
- Peter Monk, A simple proof of convergence for an edge element discretization of Maxwell’s equations, Computational electromagnetics (Kiel, 2001) Lect. Notes Comput. Sci. Eng., vol. 28, Springer, Berlin, 2003, pp. 127–141. MR 1986135, DOI 10.1007/978-3-642-55745-3_{9}
- P. Monk. Finite element methods for Maxwell’s equations. Oxford Science Pub., Oxford, 2003.
- P. Monk and L. Demkowicz, Discrete compactness and the approximation of Maxwell’s equations in ${\Bbb R}^3$, Math. Comp. 70 (2001), no. 234, 507–523. MR 1709155, DOI 10.1090/S0025-5718-00-01229-1
- J.-C. Nédélec, Mixed finite elements in $\textbf {R}^{3}$, Numer. Math. 35 (1980), no. 3, 315–341. MR 592160, DOI 10.1007/BF01396415
- J.-C. Nédélec, A new family of mixed finite elements in $\textbf {R}^3$, Numer. Math. 50 (1986), no. 1, 57–81. MR 864305, DOI 10.1007/BF01389668
- 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
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
- Received by editor(s): September 11, 2006
- Received by editor(s) in revised form: January 24, 2007
- Published electronically: September 18, 2007
- Additional Notes: This work was supported in part by the National Science Foundation through grant No. 0311902.
- © Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp. 77 (2008), 1-10
- MSC (2000): Primary 78M10, 65F10, 65N30
- DOI: https://doi.org/10.1090/S0025-5718-07-02037-6
- MathSciNet review: 2353940