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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

Abstract

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.

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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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