四阶变系数问题基于降阶格式的Legendre-Fourier谱逼近  

LEGENDRE-FOURIER SPECTRAL APPROXIMATION FOR THE FOURTH-ORDER VARIABLE COEFFICIENT PROBLEMS BASED ON THE REDUCED-ORDER SCHEMES

作　　者：杨青青 安静 Yang Qingqing;An Jing(School of Mathematics Science,Guizhou Normal University,550025,Guiyang,China)

机构地区：[1]贵州师范大学数学科学学院,贵阳550025

出　　处：《山东师范大学学报(自然科学版)》2025年第1期52-63,共12页Journal of Shandong Normal University(Natural Science Edition)

基　　金：国家自然科学基金资助项目(12461078);贵州省教育厅自然科学研究资助项目(黔教技[2023]011)。

摘　　要：针对圆域上四阶变系数问题,提出了一种基于降阶格式的Legendre-Fourier谱方法。首先,借助极坐标变换和辅助函数,将原问题转化为极坐标系下的二阶耦合系统,并根据极条件定义一类带权的Sobolev空间,建立其变分形式及其离散。然后,在适当的条件下,证明了弱解及数值解的存在性与唯一性,再结合Céa引理和非一致带权Sobolev空间中投影算子的逼近性质,弱解与数值解之间的误差估计进一步被推导。最后,通过数值实验对算法进行了验证,实验结果表明所提出的离散格式的算法设计具有谱精度。For the fourth-order variable coefficient problem on a circular domain,a spectral method based on a reduced-order scheme using Legendre-Fourier series is proposed.First,by means of polar coordinate transformation and auxiliary functions,the original problem is cast into the form of a coupled system of second-order equations in polar coordinates,and a weighted Sobolev space is defined based on the polar conditions,and its variational form and discretization are established.Then,under suitable conditions,the existence and uniqueness of both the weak and numerical solutions are demonstrated.The error estimate between the weak and numerical solutions is further derived by combining the Céa lemma and the approximation property of the projection operator in a non-uniformly weighted Sobolev space.Finally,the algorithm was validated through numerical experiments,and the results indicate that the proposed discretization scheme exhibits spectral accuracy.

关 键 词：四阶问题 降阶格式 Legendre-Fourier谱方法 误差估计 

分 类 号：O241[理学—计算数学]