Volume 12, Issue 4
Analysis of a Special $Q_1$-Finite Volume Element Scheme for Anisotropic Diffusion Problems

Numer. Math. Theor. Meth. Appl., 12 (2019), pp. 1141-1167.

Published online: 2019-06

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• Abstract

In this paper, we  analyze a special $Q_1$-finite volume element   scheme which is obtained by using the midpoint rule to approximate the line integrals in the standard $Q_1$-finite volume element method. A necessary and sufficient condition  for the positive definiteness of  the element stiffness matrix is  obtained. Based on  this result, a sufficient condition  for the coercivity of the scheme is  proposed. This sufficient condition has an explicit form  involving the information of the diffusion tensor and the mesh. In particular, this condition can reduce to a pure geometric one  that covers some special meshes, including  the parallelogram meshes,   the $h^{1+\gamma}$-parallelogram meshes and  some trapezoidal meshes. Moreover, the $H^1$ error estimate is proved rigorously without the $h^{1+\gamma}$-parallelogram assumption required by existing works. Numerical results are also presented to validate the   theoretical analysis.

• Keywords

$Q_1$-finite volume element scheme, midpoint rule, coercivity, $H^1$ error estimates.

65M10, 78A48

fangf512@nenu.edu.cn (Fang Fang)

hq1162377655@163.com (Qi Hong)

wu_jiming@iapcm.ac.cn (Jiming Wu)

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@Article{NMTMA-12-1141, author = {Fang , Fang and Hong , Qi and Wu , Jiming }, title = {Analysis of a Special $Q_1$-Finite Volume Element Scheme for Anisotropic Diffusion Problems}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2019}, volume = {12}, number = {4}, pages = {1141--1167}, abstract = {

In this paper, we  analyze a special $Q_1$-finite volume element   scheme which is obtained by using the midpoint rule to approximate the line integrals in the standard $Q_1$-finite volume element method. A necessary and sufficient condition  for the positive definiteness of  the element stiffness matrix is  obtained. Based on  this result, a sufficient condition  for the coercivity of the scheme is  proposed. This sufficient condition has an explicit form  involving the information of the diffusion tensor and the mesh. In particular, this condition can reduce to a pure geometric one  that covers some special meshes, including  the parallelogram meshes,   the $h^{1+\gamma}$-parallelogram meshes and  some trapezoidal meshes. Moreover, the $H^1$ error estimate is proved rigorously without the $h^{1+\gamma}$-parallelogram assumption required by existing works. Numerical results are also presented to validate the   theoretical analysis.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.OA-2018-0080}, url = {http://global-sci.org/intro/article_detail/nmtma/13218.html} }
TY - JOUR T1 - Analysis of a Special $Q_1$-Finite Volume Element Scheme for Anisotropic Diffusion Problems AU - Fang , Fang AU - Hong , Qi AU - Wu , Jiming JO - Numerical Mathematics: Theory, Methods and Applications VL - 4 SP - 1141 EP - 1167 PY - 2019 DA - 2019/06 SN - 12 DO - http://dor.org/10.4208/nmtma.OA-2018-0080 UR - https://global-sci.org/intro/nmtma/13218.html KW - $Q_1$-finite volume element scheme, midpoint rule, coercivity, $H^1$ error estimates. AB -

In this paper, we  analyze a special $Q_1$-finite volume element   scheme which is obtained by using the midpoint rule to approximate the line integrals in the standard $Q_1$-finite volume element method. A necessary and sufficient condition  for the positive definiteness of  the element stiffness matrix is  obtained. Based on  this result, a sufficient condition  for the coercivity of the scheme is  proposed. This sufficient condition has an explicit form  involving the information of the diffusion tensor and the mesh. In particular, this condition can reduce to a pure geometric one  that covers some special meshes, including  the parallelogram meshes,   the $h^{1+\gamma}$-parallelogram meshes and  some trapezoidal meshes. Moreover, the $H^1$ error estimate is proved rigorously without the $h^{1+\gamma}$-parallelogram assumption required by existing works. Numerical results are also presented to validate the   theoretical analysis.

Fang Fang, Qi Hong & Jiming Wu. (2019). Analysis of a Special $Q_1$-Finite Volume Element Scheme for Anisotropic Diffusion Problems. Numerical Mathematics: Theory, Methods and Applications. 12 (4). 1141-1167. doi:10.4208/nmtma.OA-2018-0080
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