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The gradient superconvergence of the finite volume method for a nonlinear elliptic problem of nonmonotone type

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Abstract

We study the superconvergence of the finite volume method for a nonlinear elliptic problem using linear trial functions. Under the condition of C-uniform meshes, we first establish a superclose weak estimate for the bilinear form of the finite volume method. Then, we prove that on the mesh point set S, the gradient approximation possesses the superconvergence: \({\max _{P \in S}}|(\nabla u - \overline \nabla {u_h})(P)| = O({h^2})|\ln h{|^{3/2}}\), where \(\overline \nabla \) denotes the average gradient on elements containing vertex P. Furthermore, by using the interpolation post-processing technique, we also derive a global superconvergence estimate in the H 1-norm and establish an asymptotically exact a posteriori error estimator for the error ‖u − u h ‖1.

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Authors and Affiliations

  1. Department of Mathematics, State Key Laboratory of Synthetical Automation for Process Industries, Northeastern University, Shenyang, 110004, China

    Tie Zhang

  2. Research Center for Mathematics and Economics, Tianjin University of Finance and Economics, Tianjin, 300222, China

    Shuhua Zhang

Authors
  1. Tie Zhang
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  2. Shuhua Zhang
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Correspondence to Shuhua Zhang.

Additional information

Dedicated to Professor Ivo Babuška on his 90th birthday

This project was supported in part by the National Basic Research Program (2012CB 955804), the Major Research Plan of the National Natural Science Foundation of China (91430108), the National Natural Science Foundation of China (11371081 and 11171251), the State Key Laboratory of Synthetical Automation for Process Industries Fundamental Research Funds (2013ZCX02), and the Major Program of Tianjin University of Finance and Economics (ZD1302).

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Zhang, T., Zhang, S. The gradient superconvergence of the finite volume method for a nonlinear elliptic problem of nonmonotone type. Appl Math 60, 573–596 (2015). https://doi.org/10.1007/s10492-015-0112-8

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  • DOI: https://doi.org/10.1007/s10492-015-0112-8

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