Korean J. Math. Vol. 26 No. 3 (2018) pp.467-481
DOI: https://doi.org/10.11568/kjm.2018.26.3.467

Postprocessing for the Raviart--Thomas mixed finite element approximation of the eigenvalue problem

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Published: Sep 30, 2018
Keywords:
eigenvalue problem, mixed finite element method, super- convergence, postprocessing, asymptotic exactness.
Mathematics Subject Classification:
65N25, 65N30.

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Kwang-Yeon Kim
Department of Mathematics Kangwon National University Chun-Cheon 24341, Korea

Abstract

In this paper we present a postprocessing scheme for the Raviart--Thomas mixed finite element approximation of the second order elliptic eigenvalue problem. This scheme is carried out by solving a primal source problem on a higher order space, and thereby can improve the convergence rate of the eigenfunction and eigenvalue approximations. It is also used to compute a posteriori error estimates which are asymptotically exact for the $L^2$ errors of the eigenfunctions. Some numerical results are provided to confirm the theoretical results.


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Supporting Agencies

This study is supported by 2015 Research Grant from Kangwon National Univer- sity (No. D1000412-01-01).

References

[1] A. Alonso., A. D. Russo, and V. Vampa, A posteriori error estimates in finite element acoustic analysis, J. Comput. Appl. Math. 117 (2000), 105–119. Google Scholar

[2] I. Babuska and J. Osborn, Eigenvalue Problems, in Handbook of Numerical Analysis II, Finite Element Methods (Part 1), edited by P.G. Lions and P.G. Ciarlet, North-Holland, Amsterdam, 1991, 641–787. Google Scholar

[3] D. Boffi, Finite element approximation of eigenvalue problems, Acta Numer. 19 (2010), 1–120. Google Scholar

[4] D. Boffi, F. Brezzi, and L. Gastaldi, On the convergence of eigenvalues for mixed formulations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 25 (1997), 131–154 Google Scholar

[5] D. Boffi, F. Brezzi, and L. Gastaldi, On the problem of spurious eigenvalues in the approximation of linear elliptic problems in mixed form, Math. Comp. 69 (2000), 121–140. Google Scholar

[6] F. Brezzi and M. Fortin, Mixed and hybrid finite element methods, Springer–Verlag, New York, 1991. Google Scholar

[7] H. Chen, S. Jia, and H. Xie, Postprocessing and higher order convergence for the mixed finite element approximations of the eigenvalue problem, Appl. Numer. Math. 61 (2011), 615–629. Google Scholar

[8] R. Dur an, L. Gastaldi, and C. Padra, A posteriori error estimators for mixed approximations of eigenvalue problems, Math. Models Methods Appl. Sci. 9 (1999), 1165-1178. Google Scholar

[9] F. Gardini, Mixed approximation of eigenvalue problems: a superconvergence result, ESAIM: M2AN 43 (2009), 853–865. Google Scholar

[10] P. Grisvard, Elliptic Problems in Non-Smooth Domains, Monographs and Studies in Mathematics 24, Pitman, Boston, 1985. Google Scholar

[11] S. Jia, H. Chen, and H. Xie, A posteriori error estimator for eigenvalue problems by mixed finite element method, Sci. China Math. 56 (2013), 887–900. Google Scholar

[12] J. Douglas Jr. and J. E. Roberts, Global estimates for mixed methods for second order elliptic equations, Math. Comp. 44 (1985), 39–52. Google Scholar

[13] Q. Lin and H. Xie, A superconvergence result for mixed finite element approximations of the eigenvalue problem, ESAIM: M2AN 46 (2012), 797–812. Google Scholar

[14] B. Mercier, J. Osborn, J. Rappaz, and P. A. Raviart, Eigenvalue approximation by mixed and hybrid methods, Math. Comp. 36 (1981), 427–453. Google Scholar

[15] A. Naga and Z. Zhang, Function value recovery and its application in eigenvalue problems, SIAM J. Numer. Anal. 50 (2012), 272–286. Google Scholar

[16] M. R. Racheva and A. B. Andreev, Superconvergence postprocessing for eigenvalues, Comp. Methods Appl. Math. 2 (2002), 171–185. Google Scholar

[17] J. Xu and A. Zhou, A two-grid discretization scheme for eigenvalue problems, Math. Comp. 70 (2001), 17–25. Google Scholar