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Superconvergence of finite volume element method for elliptic problems

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Abstract

We study the superconvergence of finite volume element (FVE) method for elliptic problems by using linear trial functions. Under the condition of C-uniform meshes, we first establish a superclose weak estimate for the bilinear form of FVE method. Then, we prove that all interior mesh points are the optimal stress points of interpolation function and further we give the superconvergence result of gradient approximation: \(\displaystyle {\max _{P\in S}}\left |\left (\nabla u-\overline {\nabla }u_{h}\right )(P)\right |=O\left (h^{2}\right )\left |\ln h\right |\), where S is the set of mesh points and \(\overline {\nabla }\) denotes the average gradient on elements containing vertex P.

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Authors and Affiliations

  1. Department of Mathematics, Northeastern University, Shenyang, 110004, People’s Republic of China

    Tie Zhang

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  1. Tie Zhang
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Correspondence to Tie Zhang.

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Communicated by: Zhongying Chen

This work was supported by the National Natural Science Funds of China, No. 11071033

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Zhang, T. Superconvergence of finite volume element method for elliptic problems. Adv Comput Math 40, 399–413 (2014). https://doi.org/10.1007/s10444-013-9313-4

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  • DOI: https://doi.org/10.1007/s10444-013-9313-4

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