The Hessian discretisation method (HDM) for fourth order linear elliptic equations provides a unified convergence analysis framework based on three properties namely coercivity, consistency, and limit-conformity. Some examples that fit in this approach include conforming and nonconforming finite element methods, finite volume methods and methods based on gradient recovery operators. A generic error estimate has been established in $L^2$, $H^1$ and $H^2$-like norms in literature. In this paper, we establish improved $L^2$ and $H^1$ error estimates in the framework of HDM and illustrate it on various schemes. Since an improved $L^2$ estimate is not expected in general for finite volume method (FVM), a modified FVM is designed by changing the quadrature of the source term and a superconvergence result is proved for this modified FVM. In addition to the Adini nonconforming finite element method (ncFEM), in this paper, we show that the Morley ncFEM is an example of HDM. Numerical results that justify the theoretical results are also presented.

中文翻译：

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改进了 Hessian 离散化方法的 L 2 和 H 1 误差估计

用于四阶线性椭圆方程的 Hessian 离散化方法 (HDM) 提供了一个基于三个属性的统一收敛分析框架，即矫顽力、一致性和极限一致性。适合这种方法的一些示例包括符合和非符合有限元方法、有限体积方法和基于梯度恢复算子的方法。已经在文献中的 $L^2$、$H^1$ 和 $H^2$-like 规范中建立了通用误差估计。在本文中，我们在 HDM 的框架中建立了改进的 $L^2$ 和 $H^1$ 误差估计，并在各种方案中对其进行了说明。由于对于有限体积法 (FVM) 通常不期望改进的 $L^2$ 估计值，因此通过改变源项的正交设计了改进的 FVM，并证明了该改进的 FVM 的超收敛结果。除了 Adini 非一致性有限元方法 (ncFEM)，在本文中，我们还展示了 Morley ncFEM 是 HDM 的一个示例。还提供了证明理论结果合理的数值结果。