Guaranteed Error Bounds for Conforming Approximations of a Maxwell Type Problem

  • Pekka NeittaanmäkiEmail author
  • Sergey Repin
Part of the Computational Methods in Applied Sciences book series (COMPUTMETHODS, volume 15)


This paper is concerned with computable error estimates for approximations to a boundary-value problem
$$\mathrm{curl}\ {\mu }^{-1}\mathrm{curl}\ u + {\kappa }^{2}u = j\quad \textrm{ in }\Omega ,$$
where μ > 0 and κ are bounded functions. We derive a posteriori error estimates valid for any conforming approximations of the considered problems. For this purpose, we apply a new approach that is based on certain transformations of the basic integral identity. The consistency of the derived a posteriori error estimates is proved and the corresponding computational strategies are discussed.

Key words

A posteriori estimates the Maxwell equation guaranteed bounds of approximation errors 


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  1. 1.
    R. Beck, R. Hiptmair, R. H. W. Hoppe, and B. Wohlmuth. Residual based a posteriori error estimators for eddy current computation. M2AN Math. Model. Numer. Anal., 34(1):159–182, 2000.zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    D. Braess and Schöberl. Equilibrated residual error estimator for Maxvell’s equation. To appear.Google Scholar
  3. 3.
    G. Duvaut and J.-L. Lions. Les inéquations en mécanique et en physique. Dunod, Paris, 1972.zbMATHGoogle Scholar
  4. 4.
    V. Girault and P.-A. Raviart. Finite element methods for Navier–Stokes equations. Theory and algorithms. Springer, Berlin, 1986.Google Scholar
  5. 5.
    G. Haase, M. Kuhn, and U. Langer. Parallel multigrid 3D Maxwell solvers. Parallel Comput., 27(6):761–775, 2001.zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    R. Hiptmair. Multigrid method for Maxwell’s equations. SIAM J. Numer. Anal., 36(1):204–225, 1999.CrossRefMathSciNetGoogle Scholar
  7. 7.
    P. Houston, I. Perugia, and D. Schötzau. An a posteriori error indicator for discontinuous Galerkin discretizations of H(curl)-elliptic partial differential equations. IMA J. Numer. Anal., 27(1):122–150, 2007.zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    P. Monk. Finite element methods for Maxwell’s equations. Oxford University Press, New York, 2003.zbMATHCrossRefGoogle Scholar
  9. 9.
    J.-C. Nédélec. A new family of mixed finite elements in R 3. Numer. Math., 50(1):57–81, 1986.zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    P. Neittaanmäki and S. Repin. Reliable methods for computer simulation. Error control and a posteriori estimates. Elsevier, Amsterdam, 2004.Google Scholar
  11. 11.
    O. Pironneau. Computer solutions of Maxwell’s equations in homogeneous media. Internat. J. Numer. Methods Fluids, 43(8):823–838, 2003. ECCOMAS Computational Fluid Dynamics Conference, Part III (Swansea, 2001).zbMATHMathSciNetADSGoogle Scholar
  12. 12.
    S. Repin. A posteriori error estimation for nonlinear variational problems by duality theory. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 243:201–214, 1997. Translation in J. Math. Sci. (New York) 99(1):927–935, 2000.Google Scholar
  13. 13.
    S. Repin. A posteriori error estimation for variational problems with uniformly convex functionals. Math. Comp., 69(230):481–500, 2000.zbMATHCrossRefMathSciNetADSGoogle Scholar
  14. 14.
    S. Repin. On the derivation of functional a posteriori estimates from integral identities. In W. Fitzgibbon, R. Hoppe, J. Periaux, O. Pironneau, and Y. Vassilevski, editors, Advances in Numerical Mathematics. Proceedings of International Conference on the Occasion of the 60th birthday of Yu. A. Kuznetsov (Moscow, 2005), pages 217–242, Moscow and Houston, TX, 2006. Institute of Numerical Mathematics of Russian Academy of Sciences and Department of Mathematics, University of Houston.Google Scholar
  15. 15.
    S. Repin. Functional a posteriori estimates for Maxwell’s equation. J. Math. Sci. (N. Y.), 142(1):1821–1827, 2007.CrossRefMathSciNetzbMATHGoogle Scholar
  16. 16.
    S. Repin. A posteriori error estimation methods for partial differential equations. In M. Kraus and U. Langer, editors, Lectures on Advanced Computational Methods in Mechanics, pages 161–226. Walter de Gruyter, Berlin, 2007.Google Scholar
  17. 17.
    S. Repin and S. Sauter. Functional a posteriori estimates for the reaction-diffusion problem. C. R. Math. Acad. Sci. Paris, 343(5):349–354, 2006.zbMATHMathSciNetGoogle Scholar
  18. 18.
    J. Saranen. On an inequality of Friedrichs. Math. Scand., 51(2):310–322, 1982.zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  1. 1.Department of Mathematical Information TechnologyUniversity of JyväskyläJyväskyläFinland
  2. 2.Department of Applied MathematicsSt. Petersburg State Technical UniversitySt. PetersburgRussia

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