Steklov-Lame特征值问题自适应多网格方法的后验误差估计.pdf

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 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 approximate 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-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 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+180Steklov-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0IntroductionWhen 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 this 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 discretization 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 National 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 Hai（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 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 indicators.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,quantum 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 adaptive 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-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 section 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 article,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 bounded 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)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 being 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 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 Eigenproblem159where=+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 kkais 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)RHhsuch 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 problems(4)and(5)have a uniquesolution,respectively.Thus,we can define the solution operators as follows:For f L2(),define A:L2()H1()satisfyinga(Af,v)=b(f,v),v H1(),and define Ah:L2()Hh H1()satisfyinga(Ahf,v)=b(f,v),vHh.And define the operator T:L2()L2()satisfying T f=(Af),and the operator Th:L2()Hh L2()satisfyingAhf=(Thf),whereand represent the restriction on.Then(2)and(3)have the following equivalent operator forms,respectivelyAu=1u,Ahuh=1huh.For(4),the following regularity estimate is given in reference 3.Remark 1Let w be the solution of the problem(4).Then for any f L2(),w H1+r(),r 1/2 and can bearbitrarily close to 1/2,andkwk1+r,+kdivwkr,.kfk0,.Define the projection operator Ph:H1()Hhbya(wPhw,vh)=0,vhHh(6)Then Ah=PhA.160Journal of Xinjiang University(Natural Science Edition in Chinese and English)2024Denote(h)=supfL2(),kfk0,=1infvHhkAf vka.Lemma 1Let w be the solution of(4).If w H1+r()(01,the first term on the right-hand side of the above inequality is the dominant term while the other two terms arehigher-order terms,thus(32)holds.Also,from Lemma 8,(18),and Lemma 3,we haveXKThl2hl(u,K).XKThlkb ek21,K+Xebkjujb j,hluk20,e.kb ek2a+|jb j,hl|2+kujuk20,.kb ek2a+4hl(j)+(hl)22hl(j),therefore,hl(u,).kujuhljka+kuhljuka+(hl)hl(j).By using Lemmas 6 and 9 again,we obtainhl(uhlj,).hl(u,)+2hl(j)kuka+kuhljuka.kujuhljka+kuhljuka+(hl)hl(j)+2hl(j)kuka.kujuhljka+hl(j)3/tl+(hl)hl(j)+2hl(j)kuka.Similarly,the first term on the right-hand side of the above inequality is the dominant term,and the remaining three terms areof higher order,thus,(33)holds.This ends the proof.Theorem 2Suppose that the conditions of Lemma 5 are satisfied,then there exists uj M(j)such that(i)For any K Thl,if K=,thenhl(uhlj,K).kujuhljk1,K+kuhljuk1,K+2hl(j)kuk1,K(34)166Journal of Xinjiang University(Natural Science Edition in Chinese and English)2024(ii)For any K Thl,if K,thenhl(uhlj,K).kujuhljk1,K+kuhljuk1,K+2hl(j)kuk1,K+XeKbkjujb j,hluk0,e(35)ProofLet uj M(j)be given by(17).By using the triangle inequality,we obtainkujuk1,Kkujuhljk1,K+kuhljuk1,K.Using(28),we obtainhl(u,K).kujuhljk1,K+kuhljuk1,K,which together with(30)yields(34).Similarly,we can prove(35).This ends the proof.From Lemma 4,it is easy to prove|hljj|=O(kujuhljk2a),and combined with Theorem 1,we obtain the followingestimation for approximate eigenvalues.Theorem 3Suppose that the conditions of Lemma 5 are satisfied,then2hl(uhlj,).|hljj|.2hl(uhlj,).Theorem 3 shows that 2hl(uhlj,)is an effective and reliable estimator of hlj.3Adaptive Multigrid Algorithm and Numerical ExperimentsReferring to the standard adaptive procedure9,1719,we utilize the error indicators given in(21)and(22)and Scheme 1to design the following adaptive multigrid algorithm based on shifted-inverse iteration.Algorithm 1Choose the parameter 01.Step 1:Select any initial grid Th1.Step 2:Solve(3)in Hh1for discrete solution(j,h1,uj,h1).Step 3:Let l=1.uh1juj,h1,h1jj,h1.Step 4:Calculate the local indicator hl(uhlj,K).Step 5:ConstructbThl Thlby Marking Strategy and parameter.Step 6:Use the program REFINE to encrypt Thlto obtain a new mesh Thl+1.Step 7:Find uHhl+1such thata(u,)hljb(u,)=b(uhlj,),Hhl+1.Denote uhl+1j=u/kukaand calculate the Rayleigh quotient hl+1j=a(uhl+1j,uhl+1j)/b(uhl+1j,uhl+1j).Step 8:Let l=l+1 and go to Step 4.Marking StrategyGive the parameter 01.Step 1:Select some elements in Thlto construct a minimal subsetbThlof Thlsuch thatXKbThl2hl(uhlj,K)XKThl2hl(uhlj,K).Step 2:Mark all elements inbThl.To verify the efficiency of Algorithm 1,we use it to solve(3)in S=(0,1)2and L=(1,1)20,1)2,respectively.Wecarry out the numerical experiments on MATLAB 2023b on a ThinkBook 14p Gen 2 PC with 16 G memory,and our programis implemented using the software package iFEM20.In our tables,the symbol hjrepresents the jth eigenvalue calculated by Algorithm 1,hjrepresents the square of theglobal error indicator h(uhj,)for the jth eigenfunction,and time represents the CPU time from the beginning of Algorithm1 to the end of the lth iteration.No.2XU Liangkun，et al：A Posteriori Error Estimation of Adaptive Multigrid Method for Steklov-Lam e Eigenproblem167Example 1Adaptive computing.We use Algorithm 1 with the conforming P1 and P2 elements to calculate the firstfour non-zero eigenvalues of(3)with =1,=1,p=1 in Sand L,respectively.We list the approximate eigenvaluesas well as the corresponding degrees of freedom Ndof and the square of the global error indicator in Tables 1 and 2.Wealso show some adaptively refined meshes in Figs 1 and 2,and depict the error curves of approximate eigenvalues and the aposteriori error indicators in Figs 3 and 4,where the reference values are taken as the most accurate approximation that wecan compute.Table 1 The results of Example 1 in Sby Algorithm 1P1P2jNdofhjhjtime/sNdofhjhjtime/s11 3542.557 270 001.223 61010.909 81 1022.532 742 167.079 71020.960 4114 0142.533 541 381.298 31021.777 213 2022.530 976 666.935 31041.917 6142 7022.531 839 573.897 31033.004 032 2262.530 966 541.178 81042.884 711 446 1182.530 986 761.060 0104118.512 01 270 0262.530 964 156.630 010889.122 721 1722.705 361 851.362 91010.417 91 0662.676 022 969.927 91020.981 1214 3242.676 390 071.202 11021.672 114 4382.673 801 676.033 71042.131 7238 1502.674 799 434.423 51032.951 833 9702.673 791 741.168 11043.191 721 698 0062.673 806 297.440 0105148.398 21 374 3462.673 789 396.360 0108157.314 741 3063.715 117 771.281 21020.687 71 2423.711 228 644.857 21030.305 1413 2323.711 496 281.187 41031.900 512 1143.711 132 283.380 01051.269 64210 8223.711 153 897.140 010513.755 7172 4663.711 131 351.550 010711.015 941 411 3543.711 134 701.100 0105111.595 71 335 0023.711 131 352.550 0109187.255 0Table 2 The results of Example 1 in Lby Algorithm 1P1P2jNdofhjhjtime/sNdofhjhjtime/s11 7841.158 384 472.517 91020.440 61 7341.156 147 766.484 01020.995 616 1281.155 832 806.444 01030.930 85 0741.155 175 132.099 11032.039 0110 1421.155 543 463.756 21031.202 69 1341.155 160 686.592 81042.435 911 714 6041.155 155 272.080 0105152.429 91 624 8181.155 153 252.070 0108185.461 723 0281.720 115 973.929 71020.306 42 9821.714 532 179.477 21030.652 6240 5881.714 806 343.036 01032.170 935 9181.714 400 097.320 01052.570 0270 2101.714 624 271.558 31033.516 962 9221.714 399 502.290 01053.937 521 541 2701.714 410 247.180 0105133.068 11 393 4701.714 399 194.640 010899.321 242 1622.137 471 375.631 91020.299 01 8902.127 555 238.069 61020.415 3412 9702.127 332 518.819 01031.141 011 5742.125 770 721.608 91031.595 4469 3822.126 037 811.628 31033.714 454 0022.125 740 727.520 01054.995 441 852 3422.125 749 645.990 0105158.409 91 460 1222.125 739 159.690 0108164.489 9For comparison,we also adopt the conforming P2 element to solve(3)on uniform meshes directly,and the results arelisted in Tables 3 and 4.For calculations on uniformly refined meshes,we use lto denote the number of uniform refinementson the initial mesh,Ndofto denote the degree of freedom to get the solution on the corresponding uniformly refined mesh,and timeto represent the time to obtain the solution on the uniformly refined mesh with the most degree of freedom(the firstcolumn in the table from right to left).It can be observed in Figs 3 and 4 that the curves of|jhj|and hjare parallel to the line with slope-1 when using the P1element or the line with slope-2 when using the P2 element,which indicates that our error indicator is effective and reliable,and the approximations obtained by adaptive computation achieve the optimal convergence;while for directly computing,the4th non-zero eigenvalu