一类四阶方程基于降阶格式的谱Galerkin逼近及误差估计

Spectral Galerkin Approximation and Error Estimates Based on Reduced Order Scheme for A Class of Fourth Order'Equations

本文针对一类四阶方程提出了一种基于降阶格式的有效谱Galerkin逼近.首先,引入一个辅助函数,将四阶方程化为两个耦合的二阶方程,并推导了它们的弱形式及其离散格式.其次,利用Lax-Milgram引理和非一致带权Sobolev空间中正交投影算子的逼近性质,严格地证明了弱解和逼近解的存在唯一性及它们之间的误差估计.最后,通过一些数值算例,数值结果表明该算法是收敛和高精度的.

In this paper, we propose a spectral Galerkin approximation and error estimates based on reduced order scheme for a class offourth order equations. Firstly, by introducing a auxiliary function, we transformthe original problems to two coupled second order equations,and their weak formand corresponding discrete format are also derived. Secondly, by using Lax-Milgramlemma and the approximationproperties of orthogonal projection operators in non-uniform weighted Sobolev spaces, we strictly prove the existence and uniquenessof weak solution and approximate solution and as well the error estimate. At the end, we conduct some numerical experiments,which show that the algorithmis convergent and high accurate.