Applications of Mathematics, Vol. 64, No. 4, pp. 383-396, 2019


Locally pointwise superconvergence of the tensor-product finite element in three dimensions

Jinghong Liu, Wen Liu, Qiding Zhu

Received August 10, 2018.   Published online July 1, 2019.

Abstract:  Consider a second-order elliptic boundary value problem in three dimensions with locally smooth coefficients and solution. Discuss local superconvergence estimates for the tensor-product finite element approximation on a regular family of rectangular meshes. It will be shown that, by the estimates for the discrete Green's function and discrete derivative Green's function, and the relationship of norms in the finite element space such as $L^2$-norms, $W^{1,\infty}$-norms, and negative-norms in locally smooth subsets of the domain $\Omega$, locally pointwise superconvergence occurs in function values and derivatives.
Keywords:  tensor-product finite element; local superconvergence; discrete Green's function
Classification MSC:  65N30


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Affiliations:   Jinghong Liu, Hainan Normal University, 99 Longkun South Road, Qiongshan Qu, Haikou, Hainan Province: 571158, China, e-mail: jhliu1129@sina.com; Wen Liu, Ningbo Institute of Technology, Zhejiang University, 1 Xuefu Rd, Yinzhou Qu, Ningbo, Zhejiang Province: 315100, China, e-mail: 529503918@qq.com; Qiding Zhu, Hunan Normal University, 36 Lushan Rd, Yuelu Qu, Changsha, Hunan Province: 410081, China, e-mail: qd_zhu@sina.com


 
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