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A non-overlapping domain decomposition for low-frequency time-harmonic Maxwell’s equations in unbounded domains

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Abstract

In this paper, we are concerned with a non-overlapping domain decomposition method for solving the low-frequency time-harmonic Maxwell’s equations in unbounded domains. This method can be viewed as a coupling of finite elements and boundary elements in unbounded domains, which are decomposed into two subdomains with a spherical artificial boundary. We first introduce a discretization for the coupled variational problem by combining Nédélec edge elements of the lowest order and curvilinear elements. Then we design a D-N alternating method for solving the discrete problem. In the method, one needs only to solve the finite element problem (in a bounded domain) and calculate some boundary integrations, instead of solving a boundary integral equation. It will be shown that such iterative algorithm converges with a rate independent of the mesh size.

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References

  1. Adams, R.: Sobolev Spaces. Academic, New York (1975)

    MATH  Google Scholar 

  2. Alonso, A., Valli, A.: Some remarks on the characterization of the space of tangential traces of H(curl, Ω) and the construction of an extension operator. Manuscripta Math. 89, 159–178 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  3. Alonso, A., Valli, A.: An optimal domain decomposition preconditioner for low-frequency time-harmonic Maxwell equations. Math. Comp. 68, 607–631 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bendali, A.: Numerical analysis of the exterior boundary value problem for the time-harmonic Maxwell equations by a boundary finite element method. Part 1: the continuous problem. Part 2: the discrete problem. Math. Comp. 43(167), 29–46 and 47–68 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  5. Buffa, A., Costabel, M., Sheen, D.: On traces for H(curl, Ω) in Lipshitz domains. J. Math. Anal. Appl. 276(2), 845–876 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  6. Buffa, A., Hiptmair, R.: Galerkin boundary element methods for electromagnetic scattering, on “computational methods in wave propagation.” In: Ainsworth, M. et al. (eds.) Lecture Notes in Computational Science and Engineering, vol. 31, pp. 83–124. Springer, Berlin Heidelberg New York (2003)

    Google Scholar 

  7. Buffa, A., Hiptmair, R., Von Petersdorff, T., Schwab, C.: Boundary element methods for Maxwell equations on Lipschitz domains. Numer. Math. (2002)

  8. Cessenat, M.: Mathematical methods in electromagnetism. In: Advances in Mathematics for Applied Sciences, vol. 41. World Scientific, Singapore (1996)

    Google Scholar 

  9. Colton, D., Kress, R.: Inverse acoustic and electromagnetic scattering theory. In: Applied Mathematical Sciences (2nd edn), vol. 93. Springer, Berlin Heidelberg New York (1998)

    Google Scholar 

  10. Dubois, F.: Discrete vector potential representation of a divergence free vector field in three dimensional domains: numerical analysis of a model problem. SIAM J. Numer. Anal. 27, 1103–1142 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  11. Girault, V., Raviart, P.: Finite Element Methods for Navier–Stokes Equations. Springer, Berlin Heidelberg New York (1986)

    MATH  Google Scholar 

  12. Hu, Q., Yu, De-hao: Solving singularity problems in unbounded domains by compling of natural BEM and composite grid FEM. Appl. Numer. Math. 37, 127–143 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  13. Hu, Q., Yu, De-hao: A solution method for a certain nonlinear interface problem in unbounded domains. Computing 67, 119–140 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  14. Hu, Q., Zou, J.: A non-overlapping domain decomposition method for Maxwell’s equations in three dimensions. SIAM J. Numer. Anal. 41, 1682–1708 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  15. Hu, Q., Zou, J.: Substructuring preconditioners for saddle-point problems arising from Maxwell’s equations in three dimensions. Math. Comp. 73, 35–61 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  16. Ja, Zu-peng, Wu, Ji-ming, Yu, De-hao: The coupled natural boundary-finite element method for solving 3-D exterior Helmholtz problem. Math. Numer. Sin. 23(3), 357–368 (2001)

    Google Scholar 

  17. Kuhn, M., Steinbach, O.: FEM-BEM coupling for 3d exterior magnetic field problems. In: Neittaanmäki, P., Tiihonen, T., Tarvainen, P. (eds.) ENUMATH 99 - Proceedings of the 3rd European Conference on Numerical Mathematics and Advanced Applications, Jyväskylä, Finland, 26–30 July 1999. pp. 180–187. World Scientific, Singapore (2000)

    Google Scholar 

  18. Lenoir, M.: Optimal isoparametric finite elements and error estimates for domains involving curved boundaries. SIAM J. Numer. Anal. 23, 562–580 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  19. Lebedev, N.N.: Special Functions and Their Applications. Dover, New York (1972)

    MATH  Google Scholar 

  20. Liu, Shi-shi, Liu, Shi-da: Special Function. Weather Press, Beijing, China (1988)

    Google Scholar 

  21. Monk, P.: Finite Element Methods for Maxwell’s Equations. Clarendon, Oxford (2003)

    MATH  Google Scholar 

  22. Nédélec, J.C.: Mixed finite elements in R 3. Numer. Math. 35, 315–341 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  23. Nédélec, J.C.: Acoustic and Electromagnetic Equations: Integral Representations for Harmonic Problems. Springer, Berlin Heidelberg New York (2001)

    MATH  Google Scholar 

  24. Quarteroni, A., Valli, A.: Numerical Approximation of Partial Differential Equations. Springer, Berlin Heidelberg New York (1994)

    MATH  Google Scholar 

  25. Toselli, A., Widlund, O., Wohlmuth, B.: An iterative substructuring method for Maxwell’s equations in two dimensions. Math. Comp. 70, 935–949 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  26. Yu, De-hao: Mathematical Theory of Natural Boundary Element Method. Science Press, Beijing, China (1993)

    Google Scholar 

  27. Yu, De-hao: A domain decomposition method based on natural boundary reduction over unbounded domain. Math. Numer. Sin. 16(4), 448–459 (1994)

    Google Scholar 

  28. Yu, De-hao, Xue, Wei-Min, Huang, Hong-ci: A Dirichlet–Neumann alternating method in infinite domain: algorithm and convergence analysis. Research Report, ICM-95-31, Chinese Academy of Sciences; Technical Report, Math-092, Department of Mathematics, Hong Kong Baptist University (1995)

  29. Yu, De-hao: Natural Boundary Integral Method and Its Applications. Kluwer, Boston, MA (2002)

    MATH  Google Scholar 

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Authors and Affiliations

  1. LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100080, People’s Republic of China

    Yang Liu, Qiya Hu & Dehao Yu

  2. Graduate University of Chinese Academy of Sciences, Beijing, 100049, People’s Republic of China

    Yang Liu

Authors
  1. Yang Liu
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  2. Qiya Hu
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  3. Dehao Yu
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Corresponding author

Correspondence to Yang Liu.

Additional information

Communicated by: Y. Xu.

This work of Yang Liu and Dehao Yu was subsidized by the National Basic Research Program of China under the grant G19990328,2005CB321701, and the National Natural Science Foundation of China under the grant 10531080.

The work of Qiya Hu was supported by Natural Science Foundation of China G10371129.

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Liu, Y., Hu, Q. & Yu, D. A non-overlapping domain decomposition for low-frequency time-harmonic Maxwell’s equations in unbounded domains. Adv Comput Math 28, 355–382 (2008). https://doi.org/10.1007/s10444-007-9027-6

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