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arXiv 1803.06985math.NAcs.NA
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四阶线性椭圆方程的Hessian离散化方法

The Hessian discretisation method for fourth order linear elliptic equations

Monash University（莫纳什大学）；University of Newcastle（纽卡斯尔大学）；Indian Institute of Technology Bombay（孟买印度理工学院）

Jérôme Droniou, Bishnu P. Lamichhane, Devika Shylaja

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AI总结：

本文提出一种统一框架HDM，基于四个离散元素和若干内在精度指标，用于解决四阶椭圆问题，涵盖有限元和有限体积方法，并设计了新的数值方法。

摘要：

本文提出了一种统一框架，即Hessian离散化方法（HDM），该方法基于四个离散元素（统称为Hessian离散化）和若干内在精度指标，与所考虑模型无关。通过仅使用这些内在指标，当HDM框架应用于线性四阶问题时，获得了误差估计。证明HDM涵盖了大量四阶椭圆问题的数值方法：有限元方法（包括符合和非符合）以及有限体积方法。我们还使用HDM设计了一种基于符合P₁有限元空间和梯度恢复算子的新方法。展示了该新方案和有限体积方案的数值实验结果。

英文摘要：

In this paper, we propose a unified framework, the Hessian discretisation method (HDM), which is based on four discrete elements (called altogether a Hessian discretisation) and a few intrinsic indicators of accuracy, independent of the considered model. An error estimate is obtained, using only these intrinsic indicators, when the HDM framework is applied to linear fourth order problems. It is shown that HDM encompasses a large number of numerical methods for fourth order elliptic problems: finite element methods (conforming and non-conforming) as well as finite volume methods. We also use the HDM to design a novel method, based on conforming $\mathbb{P}_1$ finite element space and gradient recovery operators. Results of numerical experiments are presented for this novel scheme and for a finite volume scheme.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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