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Volume 9, Issue 4
The Gradient Superconvergence of Bilinear Finite Volume Element for Elliptic Problems

Tie Zhang & Lixin Tang

DOI: 10.4208/nmtma.2016.m1515

Numer. Math. Theor. Meth. Appl., 9 (2016), pp. 579-594.

Published online: 2016-09

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  • Abstract

We study the gradient superconvergence of bilinear finite volume element (FVE) solving the elliptic problems. First, a superclose weak estimate is established for the bilinear form of the FVE method. Then, we prove that the gradient approximation of the FVE solution has the superconvergence property:

image.png

where image.png denotes the average gradient on elements containing point $P$ and $S$ is the set of optimal stress points composed of the mesh points, the midpoints of edges and the centers of elements.

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@Article{NMTMA-9-579, author = {Tie Zhang and Lixin Tang}, title = {The Gradient Superconvergence of Bilinear Finite Volume Element for Elliptic Problems}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2016}, volume = {9}, number = {4}, pages = {579--594}, abstract = {

We study the gradient superconvergence of bilinear finite volume element (FVE) solving the elliptic problems. First, a superclose weak estimate is established for the bilinear form of the FVE method. Then, we prove that the gradient approximation of the FVE solution has the superconvergence property:

image.png

where image.png denotes the average gradient on elements containing point $P$ and $S$ is the set of optimal stress points composed of the mesh points, the midpoints of edges and the centers of elements.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.2016.m1515}, url = {http://global-sci.org/intro/article_detail/nmtma/12390.html} }
TY - JOUR T1 - The Gradient Superconvergence of Bilinear Finite Volume Element for Elliptic Problems AU - Tie Zhang & Lixin Tang JO - Numerical Mathematics: Theory, Methods and Applications VL - 4 SP - 579 EP - 594 PY - 2016 DA - 2016/09 SN - 9 DO - http://doi.org/10.4208/nmtma.2016.m1515 UR - https://global-sci.org/intro/article_detail/nmtma/12390.html KW - AB -

We study the gradient superconvergence of bilinear finite volume element (FVE) solving the elliptic problems. First, a superclose weak estimate is established for the bilinear form of the FVE method. Then, we prove that the gradient approximation of the FVE solution has the superconvergence property:

image.png

where image.png denotes the average gradient on elements containing point $P$ and $S$ is the set of optimal stress points composed of the mesh points, the midpoints of edges and the centers of elements.

Tie Zhang and Lixin Tang. (2016). The Gradient Superconvergence of Bilinear Finite Volume Element for Elliptic Problems. Numerical Mathematics: Theory, Methods and Applications. 9 (4). 579-594. doi:10.4208/nmtma.2016.m1515
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