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://doi.org/10.4208/nmtma.OA-2018-0080
UR - https://global-sci.org/intro/article_detail/nmtma/13218.html
KW - $Q_1$-finite volume element scheme,  midpoint rule, coercivity, $H^1$ error estimates.
AB - <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>