论文标题
第四阶半线性椭圆方程的Hessian离散方法：vonKármán和Navier的应用 - Stokes Models
Hessian discretisation method for fourth order semi-linear elliptic equations: applications to the von Kármán and Navier--Stokes models
论文作者
论文摘要
本文介绍了针对三连线非线性的第四阶半线性椭圆方程的Hessian离散方法（HDM）。 HDM为几种数值方法的收敛分析提供了一个通用框架，例如，基于梯度恢复（GR）运算符的符合和不合格的有限元方法（NCFEM）和方法。 ADINI NCFEM和GR方法是一种基于分段线性函数的廉价局部重建的特定方案，首次分析了第四阶半线性椭圆方程，具有三线性非线性。四个属性，即，在HDM框架中不需要任何规律性的HDM框架中的固定性，一致性，极限符合性和紧凑性使得能够收敛分析。在应用中，讨论了应用程序中的两个重要问题，即，讨论了流函数涡度配方中的Navier-Stokes方程和板弯曲的VonKármán方程。为Morley NCFEM和GR方法提供了数值实验的结果。
This paper deals with the Hessian discretisation method (HDM) for fourth order semi-linear elliptic equations with a trilinear nonlinearity. The HDM provides a generic framework for the convergence analysis of several numerical methods, such as, the conforming and non-conforming finite element methods (ncFEMs) and methods based on gradient recovery (GR) operators. The Adini ncFEM and GR method, a specific scheme that is based on cheap, local reconstructions of higher-order derivatives from piecewise linear functions, are analysed for the first time for fourth order semi-linear elliptic equations with trilinear nonlinearity. Four properties namely, the coercivity, consistency, limit-conformity and compactness enable the convergence analysis in HDM framework that does not require any regularity of the exact solution. Two important problems in applications namely, the Navier--Stokes equations in stream function vorticity formulation and the von Kármán equations of plate bending are discussed. Results of numerical experiments are presented for the Morley ncFEM and GR method.
