The cell functional minimization scheme for the anisotropic diffusion problems on arbitrary polygonal grids
The cell functional minimization scheme for the anisotropic diffusion problems on arbitrary polygonal grids
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DOI：
10.1051/m2an/2014030
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发表时间：
2015
期刊：
影响因子：
--
通讯作者：
Li Yin;Jiming Wu;Zhiming Gao
中科院分区：
文献类型：
--
作者：
Li Yin;Jiming Wu;Zhiming Gao
针对非结构多边形网格，构造了一种基于单元泛函最小化的有限体积格式。该方案具有局部模板，允许任意扩散张量，当边缘未知数定义在边缘的中点处时，得到对称的正定扩散矩阵，并且是线性保持的，即保持线性解。在很弱的几何条件下，通过离散泛函方法得到了该格式的稳定性结果和离散H1误差估计。最后，在不同的网格类型(包括一种特殊的拼图网格)上的数值结果表明了该格式的良好性能，并验证了理论分析。
A finite volume scheme based on minimization of a certain cell functional is constructed for unstructured polygonal meshes. This new scheme has a local stencil, allows arbitrary diffusion tensors, leads to a symmetric positive definite diffusion matrix in case that edge unknowns are defined at the midpoints of edges, and is linearity-preserving, i.e. , preserves linear solutions. Under a very weak geometry condition, the stability result and discrete H 1 error estimate of the scheme is obtained through a discrete functional approach. Finally, numerical results on various mesh types (including a particular jigsaw puzzle mesh) demonstrate the good performance of the scheme and validate the theoretical analysis.