Superconvergence of the Gradient Approximation for Weak Galerkin Finite Element Methods on Rectangular Partitions
Superconvergence of the Gradient Approximation for Weak Galerkin Finite Element Methods on Rectangular Partitions
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发表时间：
2018-04
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通讯作者：
Dan Li;Chunmei Wang;Junping Wang
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作者：
Dan Li;Chunmei Wang;Junping Wang
本文给出了用矩形剖分的弱Galerkin有限元方法离散二阶椭圆型方程梯度逼近的一个超收敛结果。结果表明，对于由弱Galerkin方法得到的近似梯度，采用分段线性常值函数组成的最低阶元，其误差估计为O（h^2）阶。对于这个数值格式，梯度逼近的最佳收敛阶已被证明是O（h）。$O（h^2）$-收敛优于最优阶，从而揭示了相应的弱Galerkin有限元方法的一个上级性质.一些计算结果报告支持和说明超收敛理论的数值。
This article presents a superconvergence result for the gradient approximation of the second order elliptic equation discretized by the weak Galerkin finite element method with rectangular partitions. The result shows an error estimate of order $O(h^2)$ for the approximate gradient obtained from the weak Galerkin using the lowest order element consisting of piecewise linear and constant functions. For this numerical scheme, the optimal order of convergence for the gradient approximation has shown to be $O(h)$. The $O(h^2)$-convergence outperforms the optimal order, and therefore reveals a superior property for the corresponding weak Galerkin finite element method. Some computational results are reported to support and illustrate the superconvergence theory numerically.