Analysis of a special ${Q_1}$-finite volume element scheme for anisotropic diffusion problems
Analysis of a special ${Q_1}$-finite volume element scheme for anisotropic diffusion problems
复制标题
各向异性扩散问题的特殊Q(1)-有限体积元方案分析
DOI：
10.4208/nmtma.oa-2018-0080
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发表时间：
2019
期刊：
影响因子：
--
通讯作者：
Jiming Wu
中科院分区：
文献类型：
--
作者：
Fang Fang;Qi Hong;Jiming Wu
本文分析了一种特殊的${Q_1}$-有限体积元格式，该格式是在标准${Q_1}$-有限体积元方法中利用中点法则逼近线积分而得到的。得到了单元刚度矩阵为正定性的充分必要条件。在此基础上，给出了该方案的收敛性的一个充分条件。该充分条件具有包含扩散张量和网格信息的显式形式。特别地，这个条件可以简化为一个覆盖一些特殊网格的纯几何条件，包括三角形网格，${h^{1 + \gamma }}$-三角形网格和一些梯形网格。此外，在不需要现有工作中所要求的${h^{1 + \gamma }}$-∞假设的情况下，严格地证明了${H^1}$误差估计.数值结果也验证了理论分析。
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.