网格的各向异性对非协调旋转Q1元特征值上下界及精度的影响  

Effects of Anisotropy of Grids on both Upper and Tower Bounds and Accuracy of Eigenvalues by the Nonconforming Rotated Q1 Element

作　　者：李友爱 LI You-ai(School Mathematics and Statistics,Beijing Technology and Business University,Beijing 100048,China)

机构地区：[1]北京工商大学数学与统计学院,北京100048

出　　处：《数学的实践与认识》2020年第22期226-232,共7页Mathematics in Practice and Theory

摘　　要：对Laplace算子特征值问题,基于非协调旋转Q1元,在方形区域上,研究网格的各向异性对特征值的上下界及精度的影响.数值实验表明,在方型区域上,利用非协调旋转Q1元做近似计算时,整体各向异性网格会导致二重特征值中一个为上界逼近,另一个为下界逼近;但单重特征值的上下界逼近性与均匀网格一致.而局部各向异性网格会导致单重特征值的上下界逼近性发生改变,但二重特征值的上下界逼近性与均匀网格一致.从误差来看,均匀网格上的误差与局部各向异性网格上的误差大小相似,而整体各向异性网格上的近似误差要比均匀网格及局部各向异性网格上的误差更大,并且网格的整体各向异性越强,误差越大.还有,不管在哪种网格条件下,单重特征值的误差要比二重特征值的误差小.由此可知,对非协调元,特征值上下界逼近性及精度与问题本身、所使用单元、网格各向异性等都有关系.因此,如果原问题本身不具有各向异性,则采用各向异性网格近似意义不大.In this paper,for the Laplace operator eigenvalue problem in the square domain,based on the nonconforming rotated Q1 element,the effect of the anisotropy of meshes on the property of the upper or lower bound and the accuracy of the approximate eigenvalues is numerically studied.Numerical results show that,on global anisotropic meshes,one approximate eigenvalue is a lower bound of the exact one and the other one is an upper bound for double eigenvalues while on isotropic meshes both approximate eigenvalues are an upper bound.For single eigenvalues,the property of the upper or lower bound of the approximate eigenvalue on global anisotropic meshes is the same as that on isotropic meshes.However,on local anisotropic meshes,both approximate eigenvalues are an upper bound for double eigenvalues while compared with that on isotropic meshes,the approximate eigenvalue changes from an upper(lower) bound to a lower(upper) bound for single eigenvalues.The error on isotropic meshes is similar to that on the local anisotropic mesh,However,the error on the global anisotropic grids is larger than that on both isotropic grids and local anisotropic grids,and the stronger the global anisotropy of the grid is,the larger the error is.In addition,the error of single eigenvalues is smaller than that of double eigenvalues.Therefore,for the nonconforming element,the property of the upper or lower bound and the accuracy of approximate eigenvalues are related to the problem itself,the elements used,the anisotropy of meshes.The observation is that if the problem itself is not anisotropic,it is of little significance to adopt anisotropic meshes.

关 键 词：LAPLACE算子 特征值问题 非协调旋转Q1元 上界 下界 

分 类 号：O241.8[理学—计算数学]