A FINITE VOLUME METHOD FOR THE LAPLACE EQUATION ON ALMOST ARBITRARY TWO-DIMENSIONAL GRIDS
A FINITE VOLUME METHOD FOR THE LAPLACE EQUATION ON ALMOST ARBITRARY TWO-DIMENSIONAL GRIDS
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DOI：
10.1051/m2an:2005047
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发表时间：
2005-11
期刊：
影响因子：
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通讯作者：
K. Domelevo;P. Omnes
中科院分区：
文献类型：
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作者：
K. Domelevo;P. Omnes
本文提出了一种基于拉普拉斯方程在几乎任意二维原始网格单元和通过连接原始网格单元中心而得到的对偶网格单元上积分的有限体积法。关键成分是离散梯度和发散算子的定义，验证了离散绿色公式。这种方法推广了现有的有限体积法，需要“Voronoi型”网格。我们证明了这种有限体积法与非协调有限元法的等效性，其基函数为P1的细胞，通常称为“钻石细胞”，第三个网格。在几何条件下，这些钻石细胞，我们证明了一个一阶收敛在H1范数和L2范数。超收敛结果在某些类型的同位细化网格上得到。最后，数值实验证实了这些结果，也表明在一般网格上的L2范数的二阶收敛。它们还表明，该方法执行特别好的近似的梯度的解决方案，并可用于退化三角形网格。最后给出了在非协调局部加密网格上的应用实例.
We present a finite volume method based on the integration of the Laplace equation on both the cells of a primal almost arbitrary two-dimensional mesh and those of a dual mesh obtained by joining the centers of the cells of the primal mesh. The key ingredient is the definition of discrete gradient and divergence operators verifying a discrete Green formula. This method generalizes an existing finite volume method that requires "Voronoi-type" meshes. We show the equivalence of this finite volume method with a non-conforming finite element method with basis functions being P 1 on the cells, generally called "diamond-cells", of a third mesh. Under geometrical conditions on these diamond- cells, we prove a first-order convergence both in the H 1 norm and in the L 2 norm. Superconvergence results are obtained on certain types of homothetically refined grids. Finally, numerical experiments confirm these results and also show second-order convergence in the L 2 norm on general grids. They also indicate that this method performs particularly well for the approximation of the gradient of the solution, and may be used on degenerating triangular grids. An example of application on non- conforming locally refined grids is given.