Steklov-Lame特征值问题自适应多网格方法的后验误差估计.pdf
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1、第41卷第2期2024年3月新疆大学学报(自然科学版中英文)Journal of Xinjiang University(Natural Science Edition in Chinese and English)Vol.41,No.2Mar.,2024A Posteriori Error Estimation of Adaptive MultigridMethod for Steklov-Lam e EigenproblemXU Liangkun,BI Hai(School of Mathematical Sciences,Guizhou Normal University,Guiyang
2、 Guizhou 550025,China)Abstract:We establish a finite element multigrid discretization scheme based on the shifted-inverse iteration for the Steklov-Lam e eigenproblem,and investigate the a posteriori error estimation of residual type for the scheme.Firstly,we give the errorestimation of the approxim
3、ate eigenfunction in the sense of L2()norm,then we give the a posteriori error indicators for themultigrid approximate solution,and prove the reliability and efficiency of the indicators.Finally,we use the a posteriori errorindicators to design an adaptive multigrid algorithm for solving the Steklov
4、-Lam e eigenproblem.Key words:Steklov-Lam e eigenvalues;multigrid discretization based on shifted inverse iteration;a posteriori error estimation;adaptive multigrid algorithmDOI:10.13568/ki.651094.651316.2023.12.24.0002CLC number:O241.1Document Code:AArticle ID:2096-7675(2024)02-0157-014引文格式:徐良坤,闭海
5、Steklov-Lam e 特征值问题自适应多网格方法的后验误差估计J 新疆大学学报(自然科学版中英文),2024,41(2):157-170+180英文引文格式:XU Liangkun,BI Hai A posteriori error estimation of adaptive multigrid method for Steklov-Lam e eigenprob-lemJ Journal of Xinjiang University(Natural Science Edition in Chinese and English),2024,41(2):157-170+180Steklo
6、v-Lam e特征值问题自适应多网格方法的后验误差估计徐良坤,闭 海(贵州师范大学 数学科学学院,贵州 贵阳550025)摘要:建立 Steklov-Lam e 特征值问题的一种基于移位反迭代的有限元多网格离散方案,并研究该方案基于残差型的后验误差估计.首先给出近似特征函数在 L2()范数意义下的误差估计,其次给出多网格方案近似解的后验误差指示子,并证明后验误差指示子的可靠性和有效性.最后利用后验误差指示子设计自适应多网格算法并用于求解 Steklov-Lam e 特征值问题.关键词:Steklov-Lam e 特征值;基于移位反迭代的多网格离散;后验误差估计;自适应多网格算法0Introdu
7、ctionWhen the spectral parameter appears in the boundary conditions,we refer to this type of eigenvalue problem as aSteklov-type eigenvalue problem1.In elasticity,Dom nguez2first introduced the Steklov-Lam e eigenvalue problem inwhich the spectral parameter appears in Robin boundary conditions.For t
8、his problem,Dom nguez2explored the existenceof a countable spectrum and derived the a priori error estimates.Li et al.3proposed a discontinuous Galerkin method ofNitsches version and provided numerical experiments to show that the method is locking-free,and Xu et al.4discussed amultigrid discretizat
9、ion scheme of discontinuous Galerkin method based on the shifted-inverse iteration.As far as we know,there is no literature reporting the a posteriori error estimation of the multigrid scheme for this problem.So,the aim of Received Date:2023-12-24Foundation Item:This work was supported by the Nation
10、al Natural Science Foundation of the Peoples Republic of China“The research of finiteelement methods for eigenvalue problems in inverse scattering”(12261024)Biography:XU Liangkun(1998),male,master student,research fields:finite element method for eigenvalue problems,E-mail: Corresponding author:BI H
11、ai(1977),female,professor,research fields:finite element method for eigenvalue problems,E-mail:158Journal of Xinjiang University(Natural Science Edition in Chinese and English)2024this paper is to investigate the a posteriori error estimation of residual type of multigrid discretization scheme based
12、 on theshifted-inverse iteration for the Steklov-Lam e eigenproblem.To do so,we first give the error estimation of the approximateeigenfunction in the L2()norm.Then we give the a posteriori error indicators for the multigrid approximate solution,andprove the reliability and efficiency of the indicat
13、ors.Adaptive finite element(FE)methods have been widely used as an efficient numerical method for solving eigenvalueproblems,for example,Maxwell eigenvalue problems56,elastic eigenvalue problems78,Steklov eigenvalue problems910,transmission eigenvalue problems11,Stokes eigenvalue problems1213,quantu
14、m eigenvalue problem14,et al.The theo-retical basis of adaptive FE methods is the a posteriori error estimation.After giving the a posteriori error indicators andconducting an a posteriori error analysis,we design an adaptive multigrid algorithm based on shifted-inverse iteration.Weimplement the ada
15、ptive calculation,and the numerical results indicate that our algorithm is effective in solving Steklov-Lam eeigenproblem with constant coefficients and discontinuous constant coefficients.The remainder of this paper is organized as follows.In section 1,we present the FE approximation of the Steklov
16、-Lam eeigenproblem and derive the error estimate for the approximate eigenfunctions in the L2()norm.In section 2,we applythe multigrid scheme in 15 to the Steklov-Lam e eigenproblem.We then give the a posteriori error indicators and conduct ana posteriori error analysis for the multigrid scheme.In s
17、ection 3,we establish an adaptive multigrid algorithm based on theshifted-inverse iteration and exhibit some numerical examples to verify the efficiency and accuracy of the proposed method.Finally,in section 4,we provide a summary of the article and prospects for future research.Throughout this arti
18、cle,we use the letter C with or without subscripts to represent a generic positive constant that isindependent of the mesh size h and may take different values in different contexts,and use the notation“a.b”to meanaCb.1Conforming FE Approximation of Steklov-Lam e EigenproblemLet x=(x1,x2)T,R2be a bo
19、unded polygonal domain with Lipschitz continuous boundary,where representsthe region occupied by an isotropic elastic body.Let ndenote the unit outward normal vector on,H1():=H1()H1(),and L2():=L2()L2()(=,).The Steklov-Lam e eigenvalue problem we considered is to find u,0 and R suchthatdiv(u)=0,in(u
20、)n=pu,on(1)where p(x),u=u1(x),u2(x)T,and represent the density,displacement,and frequency of the elastic body,respectively.The Cauchy stress tensor is given by(u)=2(u)+tr(u)I.Here,IR22is the identity tensor,R and 0 are Lam ecoefficients satisfying+0,and(u)=?u+(u)T?/2 is the strain tensor with u bein
21、g the displacement gradient tensorthat is defined as followsu=u1/x1u1/x2u2/x1u2/x2.We also introduce the following symbols()i=2Xj=1ijxj,(n)i=2Xj=1ijnj,:=2Xi,j=1ijij.Suppose that pL()has a positive lower bound on.LetRM():=v H1()|v(x)=a+Bx,aR2,BR22,BT=B,x.It is easy to know that 0 is an eigenvalue of
22、the problem(1),and u RM()is the corresponding eigenvector(see 2).Tofind the nonzero eigenvalues of(1),we use the following weak form(see 1-2):Find(,u)RH1()such thata(u,v)=b(u,v),v H1()(2)No.2XU Liangkun,et al:A Posteriori Error Estimation of Adaptive Multigrid Method for Steklov-Lam e Eigenproblem15
23、9where=+1,a(u,v):=Z(u):(v)dx+Zpuvds=2 Z(u):(v)dx+Z(divu)(divv)dx+Zpuvds,u,v H1(),b(u,v):=Zpuvds,u,v H1().2 proved that a(,)is continuous and H1-elliptic,and b(,)is bounded.Without loss of generality,we assume p1 in thispaper.Let kvkb=b(v,v)1/2and kvka=a(v,v)1/2,then it is clear that kkb=kk0,and kkai
24、s equivalent to the standard normkk1,in H1().Let Thbe a regular subdivision of,hKand hebe the diameters of element K and edge e,respectively,and h=maxhK:K Th be the mesh size.Define the conforming FE spaceHh=vh H1():vh|KPk(K)Pk(K),K Th.Then the conforming FE discretization of(2)is to find(h,uh)RHhsu
25、ch thata(uh,vh)=hb(uh,vh),vhHh(3)The source problem associated with(2)is as follows:Find w H1()such thata(w,v)=b(f,v),v H1()(4)Then the conforming FE discretization of(4)is to find whHhsuch thata(wh,v)=b(f,v),vHh(5)Since a(,)is continuous and H1-elliptic,according to the Lax-Milgram theorem,the prob
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- Steklov Lame 特征值 问题 自适应 网格 方法 误差 估计
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