@Article{NMTMA-12-4,
author = {Qi, Hong and Wu, Jiming and Fang, Fang and Qi, Hong 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 = {<p style="text-align: justify;">In this paper, we&nbsp; analyze a special $Q_1$-finite volume element&nbsp; &nbsp;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&nbsp; for the positive definiteness of&nbsp; the element stiffness matrix is&nbsp; obtained. Based on&nbsp; this result, a sufficient condition&nbsp; for the coercivity of the scheme is&nbsp; proposed. This sufficient condition has an explicit form&nbsp; involving the information of the diffusion tensor and the mesh. In particular, this condition can reduce to a pure geometric one&nbsp; that covers some special meshes, including the parallelogram meshes, the $h^{1+\gamma}$-parallelogram meshes and&nbsp; 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.</p>},
issn = {2079-7338},
doi = {https://doi.org/10.4208/nmtma.OA-2018-0080},
url = {https://global-sci.com/article/90434/analysis-of-a-special-q-1-finite-volume-element-scheme-for-anisotropic-diffusion-problems}
}