In this paper, a piecewise quadratic finite element method on rectangular grids for the $H^1$ problems is presented. The proposed method can be viewed as a reduced rectangular Morley element. For the source problem, the convergence rate of this scheme is $O(h^2)$ in the energy norm on uniform grids. Besides, a lower bound of the $L^2$-norm error is also proved, which makes the capacity analysis of this scheme more clear. On the other hand, for the eigenvalue problem, the numerical eigenvalues by this element are shown to be the lower bounds of the exact ones. Some numerical results are presented, which show the potential of the proposed finite element.

中文翻译：

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用于 $$H^1$$ 问题的矩形网格上的最优二次元

在本文中，针对$H^1$问题，提出了矩形网格上的分段二次有限元方法。所提出的方法可以看作是一个简化的矩形莫雷元素。对于源问题，该方案在均匀网格上的能量范数中收敛速度为$O(h^2)$。此外，还证明了$L^2$-范数误差的下界，使得该方案的容量分析更加清晰。另一方面，对于特征值问题，该元素的数值特征值显示为精确特征值的下界。给出了一些数值结果，显示了所提出的有限元的潜力。