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

Tie Zhang & Lixin Tang

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 = {}, 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 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 & Lixin Tang. (2019). 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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