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A Priori and A Posteriori Estimates of Conforming and Mixed FEM for a Kirchhoff Equation of Elliptic Type

Asha K. Dond and Amiya K. Pani


In this article, a priori and a posteriori estimates of conforming and expanded mixed finite element methods for a Kirchhoff equation of elliptic type are derived. For the expanded mixed finite element method, a variant of Brouwer’s fixed point argument combined with a monotonicity argument yields the well-posedness of the discrete nonlinear system. Further, a use of both Helmholtz decomposition of L2-vector valued functions and the discrete Helmholtz decomposition of the Raviart–Thomas finite elements helps in a crucial way to achieve optimal a priori as well as a posteriori error bounds. For both conforming and expanded mixed form, reliable and efficient a posteriori estimators are established. Finally, the numerical experiments are performed to validate the theoretical convergence rates.

MSC 2010: 65N15; 65N30

Funding statement: The first author acknowledges the financial support of the Council of Scientific and Industrial Research (CSIR), Government of India (CSIR Sr.No. 09/087(0599)/2010-EMR-I). The second author gratefully acknowledges the research support of the Department of Science and Technology, Government of India, through the National Programme on Differential Equations: Theory, Computation and Applications, DST project no. SERB/F/1279/2011-2012.


[1] Ainsworth M. and Oden J. T., A Posteriori Error Estimation in Finite Element Analysis, Pure Appl. Math. (New York), John Wiley & Sons, New York, 2000. Search in Google Scholar

[2] Alves C. O., Correa F. J. S. A. and Ma T. F., Positive solutions for a quasilinear elliptic equation of Kirchhoff type, Comput. Math. Appl. 49 (2005), 85–93. Search in Google Scholar

[3] Arnold D. N., Falk R. S. and Winther R., Preconditioning in H(div) and applications, Math. Comp. 66 (1997), 957–984. Search in Google Scholar

[4] Braess D., Finite elements: Theory, Fast Solvers, and Applications in Solid Mechanics, Cambridge University Press, Cambridge, 2007. Search in Google Scholar

[5] Brenner S. C. and Scott L. R., The Mathematical Theory of Finite Element Methods, Springer, New York, 2008. Search in Google Scholar

[6] Brezis H., Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2011. Search in Google Scholar

[7] Brezzi F. and Fortin M., Mixed and Hybrid Finite Element Methods, Springer, New York, 1991. Search in Google Scholar

[8] Browder F. E., Nonlinear monotone operators and convex sets in Banach spaces, Bull. Amer. Math. Soc. 71 (1965), 780–785. Search in Google Scholar

[9] Carstensen C., Dond A. K., Nataraj N. and Pani A. K., Error analysis of nonconforming and mixed FEMs for second-order linear non-selfadjoint and indefinite elliptic problems, Numer. Math. 133 (2016), 557–597. Search in Google Scholar

[10] Chen Z., Expanded mixed finite element methods for linear second-order elliptic problems. I, RAIRO Modél. Math. Anal. Numér. 32 (1998), 479–499. Search in Google Scholar

[11] Chen Z., Expanded mixed finite element methods for quasilinear second order elliptic problems. II, RAIRO Modél. Math. Anal. Numér. 32 (1998), 501–520. Search in Google Scholar

[12] Chipot M. and Lovat B., Some remarks on nonlocal elliptic and parabolic problems, Nonlinear Anal. 30 (1997), 4619–4627. Search in Google Scholar

[13] Chipot M. and Rodrigues J. F., On a class of nonlocal nonlinear elliptic problems, RAIRO Modél. Math. Anal. Numér. 26 (1992), 447–467. Search in Google Scholar

[14] Chipot M., Valente V. and Caffarelli G. V., Remarks on a nonlocal problem involving the Dirichlet energy, Rend. Semin. Mat. Univ. Padova 110 (2003), 199–220. Search in Google Scholar

[15] Clément P., Approximation by finite element functions using local regularization, Rev. Franc. Automat. Inform. Rech. Operat. 9 (1975), no. R-2, 77–84. Search in Google Scholar

[16] Garralda-Guillem A. I., Ruiz Galán M., Gatica G. N. and Márquez A., A posteriori error analysis of twofold saddle point variational formulations for nonlinear boundary value problems, IMA J. Numer. Anal. 34 (2014), 326–361. Search in Google Scholar

[17] Gatica G. N., A Simple Introduction to the Mixed Finite Element Method Theory and Applications, Springer Briefs Math., Springer, Cham, 2014. Search in Google Scholar

[18] Girault V. and Raviart P., Finite Element Methods for Navier–Stokes Equations, Springer, Berlin, 1980. Search in Google Scholar

[19] Gudi T., Finite element method for a nonlocal problem of Kirchhoff type, SIAM J. Numer. Anal. 50 (2012), 657–668. Search in Google Scholar

[20] Kesavan S., Topics in Functional Analysis and Application, New Age International, New Delhi, 2008. Search in Google Scholar

[21] Kim D. and Park E. J., A posteriori error estimator for expanded mixed hybrid methods, Numer. Methods Partial Differential Equations 23 (2007), 330–349. Search in Google Scholar

[22] Kirchhoff G., Vorlesungen über Mathematische Physik: Mechanik, Teubner, Leipzig, 1876. Search in Google Scholar

[23] Ma T. F., Remarks on an elliptic equation of Kirchhoff type, Nonlinear Anal. 63 (2005), 1967–1977. Search in Google Scholar

[24] Peradze J., A numerical algorithm for nonlinear Kirchhoff string equation, Numer. Math. 102 (2005), 311–342. Search in Google Scholar

[25] Vasudeva Murthy A. S., On the string equation of Narasimha, Connected at Infinity. II, Texts Read. Math. 67, Hindustan Book, New Delhi (2013), 58–84. Search in Google Scholar

[26] Verfürth R., A posteriori error estimates for nonlinear problems. Finite element discretizations of elliptic equations, Math. Comp. 62 (1994), 445–475. Search in Google Scholar

[27] Verfürth R., A Posteriori Error Estimation Techniques for Finite Element Methods, Oxford University Press, Oxford, 2013. Search in Google Scholar

Received: 2016-10-19
Revised: 2016-11-8
Accepted: 2016-11-25
Published Online: 2017-1-10
Published in Print: 2017-4-1

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