Numer. Math. Theor. Meth. Appl., 12 (2019), pp. 1141-1167.
Published online: 2019-06
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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} }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.