简支板边界条件下四阶问题基于降阶格式的一种有效的谱Galerkin逼近.pdf

Vol.43(2023)No.5?J.of Math.(PRC)?!#$%&Galerkin()+*-,./(021436527682968;:=,021?0550025)A?B:CD6EGFIHKJ4LMNPO6Q6RPSUT;VUWYXU2V2PG_;4abPcG_edGalerkinfg.hiGjlk2mn4oGpK94q6rGswUWxX6yz46m|4U_KVGWxX,466,Lax-Milgram44q44U_lf6g4,466Hl46q6f6g6U_l6P2U626Iu4_e224.26Legendre;6P_e42262UHl22rIs6_YZIpe9,6IH;4;Z66;4G_YPK6.4,24U4Hl249262,92224UHY2G_Ybc22q62I_e;6.:TeVIWX,J;L4M4NO2Q2R,KVK,d62,224MR(2010)2,CahnHilliard?14,?59?.?UP,?!1013,?!?#?$%?!1419.&?(),?!*+?+,?-/.,1023?2023,4576 89:;?,BA CDEF7G,HJI?KJL?!M+?N6 8OP3?.QR?!,S+B1TBUWV?X.?U?!,Y?Z207?A?8?OUSteklov?#S?_a?!,Y?Z217?A?$b6P8Oc?d_P?ef?!,Y?Z227?Ag?68?O?_P?aGalerkine?f.Q?$?b6P8OU?!j+kml,n?op?q+?r?sA.tu,?U!Y?Z-v,w tY?Z247?AStokes?_yx?z?!,Y?Z257U?C1dW|!,S?/?I,?JBUyV.q,B?$I7-?J?!J;Im?.?),v?|?JU/?!g?.Y;|/?7?G+m?maGalerkini?2022-07-18i:2022-09-07?:R.S,H7g?mSobolevm,?RGJm,Lax-Milgram,ef+ypm.t?,yU?SobolevJ,?S?ef?,gef+?.,Legendre?+?7,J?%42u=f(x),x D,u=(x),x D,u=(x),x D,(1.1)S&x Rd(d=2,3),DRd&J68,D(?)68Dg|.Y*+,-:p.2/0G?1Rm#S2.p?.3/,?BWe?f+?+.p?.4/,?3!4?5?6!Ga?7!8?.p?.5/?97?$!.p?.6/?97?A?$?+:?;.2=?BADCEGFIHJIKLMNPOQJmJHm/J2,/?J?SR1.1Tmmm?R,U/Sobolev#S?e?f,?R?.Vw(x)=u(x).W?(1.1)XZYRP:(w(x)=f(x),x D,w(x)=(x),x D,(2.1)(u(x)=w(x),x D,u(x)=(x),x D.(2.2)g?HISobolev:L2(D):=p:ZDp2dx ;H10(D)=p:p L2(D),xip L2(D),p|D=0,S?!:(p,q)=ZDpqdx,kpk=p(p,p);(p,q)1:=ZDpq+p qdx,kpk1=p(p,p)1.No.5_acbad:ecfagahjickalnmporqtsvuawjxpyzq|npraanprGalerkina435o?N,Y?#?$(x)=(x)=0,D=(1,1)d(d=2,3).?X)?(2.1)-(2.2)p?E/,?Be?f-gpJ#5?+.1a(w,v)pH10(D)H10(D)O?!0,|a(w,v)|kwk1kvk1,a(w,w)12kwk21.Cauchy-Schwarz!?|a(w,v)|=|ZDOwOvdx|ZD|OwOv|dx(ZD|Ow|2dx)12(ZD|Ov|2dx)12=|w|1|v|1 kwk1 kvk1.?,|w|2=|Zxi1xiwdxi|2(xi+1)Z11(xiw)2dxi,0?BD!?!ZD|w|2dx Z11(xi+1)dxiZD(xiw)2dx=2ZD(xiw)2dx.?!dkwk2 2|w|21.!?a(w,w)=ZD|Ow|2dx=|w|21d2+dkwk2112kwk21.2f L2(D),F(v)H10(D)?!0.|F(v)|.kvk1.436Vol.43v H10(D),Schwarz!?|F(v)|=|ZDfvdx|ZD|fv|dx (ZDf2dx)12(ZDv2dx)12.kvk1.Z1,2Lax-Milgram?!.1f L2(D),?(2.3)-(2.4)?Z(w,u)H10(D)H10(D),(2.5)-(2.6)?Z(wN,uN)XN XN.3wwN!?(2.3)?(2.5),!|w wN|1infvNXN|w vN|1.(2.3)(2.5),!?a(w,vN)=F(vN),vN XN,(3.1)a(wN,vN)=F(vN),vN XN.(3.2)(3.1)?(3.2)a(w wN,vN)=0,vN XN.(3.3)(3.3)Schwarz!?|w wN|21=a(w wN,w wN)=a(w wN,w vN+vN wN)=a(w wN,w vN)+a(w wN,vN wN)=a(w wN,w vN)|w wN|1|w vN|1.|w wN|1|w vN|1.vN!|w wN|1infvNXN|w vN|1.!?J,n?!?I:=(1,1)?,(x)=(1 t)(1+t)JacobiZ?,Z11J,n(t)J,m(t),(t)dt=t,nmn,t,n=kJ,nk2,(,1).?(l,k),Jacobi!26,27:Jl,kn(x)=(1 x)l(1+x)kJl,knn0(x),l,k 1,(1 x)lJl,knn0(x),l 1,k 1,(1+x)kJl,knn0(x),l 1,k 1,No.5aca:caajc?nm?Galerkin?437n n0,n0=(l+k),l,k 1,l,l 1,k 1,k,l 1,k 1.Jl,kn(t)JacobiZ?,Z11Jl,kn(t)Jl,km(t)l,k(t)dt=mn.!d!#%$?JacobiZ?:Jl,kn(x)=dYi=1Jli,kini(xi),l,k(x)=dYi=1li,ki(xi),n=(n1,n2,.,nd),l=(l1,l2,.,ld),k=(k1,k2,.,kd).!d!N&?Z?#:Q-1,-1N:=spanJ-1,-1n(x):|n|N,|n|=max1idni.d!Jacobi0)(+*,XN.-:XN=Q-1,-1N.)/)0+12:T-1,-1N:L2-1,-1(Id)Q-1,-1N,3ZId(T-1,-1Nw w)vN-1,-1dx=0,vN Q-1,-1N.(3.4)!d!#4Sobolev,Bsl,k(Id):=w:mxw L2l+s,k+s(Id),0|m|1 s,5 6879!9!:|w|Bsl,k(Id):=(dXi=1ksxiwk2l+sei,k+sei,Id)12,kwkBsl,k(Id):=(X0|m|1skmxwk2l+m,k+m,Id)12,ei:Rd#,A$,|m|1=dPi=1mi,mxw=m1x1m2x2.mdxdw.CB?D288.1E F8.14!?.4w Bs-1,-1(Id)1 s N+1!?%G H|T-1,-1Nw w|B1-1,-1(Id)Cs(N s)!(N 1)!(N+s)1s2|w|Bs-1,-1(Id),438Vol.43,IN?1J,C 2.2wwN?(2.3)?!(2.5),KIw Bs-1,-1(Id)H10(Id)1 s N+1J,MLN%G H|w wN|1.N1s|w|Bs-1,-1(Id).Z3-|w wN|1infvNXN|w vN|1.!|w wN|21infvNXN|w vN|21|w T-1,-1Nw|21.(3.5)|w T-1,-1Nw|1=nZIddXj=1?xj(w T-1,-1Nw)?2dxo12nZIddXj=1?xj(w T-1,-1Nw)?2dYi=1,i6=j1(1 xi)(1+xi)dxo12=nZIddXj=1?xj(w T-1,-1Nw)?2l+ej,k+ejo12=|w T-1,-1Nw|B1-1,-1(Id).Z4B#D28(3.5.32)!|w wN|1.s(N s)!(N 1)!(N+s)1s2|w|Bs-1,-1(Id).N1s|w|Bs-1,-1(Id).3uuN?(2.4)?(2.6),.Iu Bs-1,-1(Id)H10(Id)1 s N+1J,MLN%G H|u uN|1.N1s(|w|Bs-1,-1(Id)+|u|Bs-1,-1(Id).(2.4)(2.6)-:a(u,hN)=b(w,hN),hN XN,(3.6)a(uN,hN)=b(wN,hN),hN XN.(3.7)(3.6)?(3.7)!:a(u uN,hN)=b(w wN,hN),hN XN.(3.8)No.5aca:caajc?nm?Galerkin?439qN XN,Z1(3.8)ku uNk21 2a(u uN,u uN)=2a(u uN,u qN+qN uN)=2a(u uN,u qN)+a(u uN,qN uN)=2a(u uN,u qN)+b(w wN,qN uN)2ku uNk1ku qNk1+2kw wNkkqN uNk.MOqN=T-1,-1Nu,!ku uNk21 2ku uNk1ku T-1,-1Nuk1+2kw wNkkT-1,-1Nu uNk.PkT-1,-1Nu uNk=kT-1,-1Nu u+u uNk kT-1,-1Nu uk1+ku uNk1,2kw wNkkT-1,-1Nu uk1 kw wNk2+kT-1,-1Nu uk21,2kw wNkku uNk1 4kw wNk2+14ku uNk21,2ku uNk1ku T-1,-1Nuk114ku uNk21+4ku T-1,-1Nuk21,Poincar e!?,4?2ku uNk21 10kw wNk2+10ku T-1,-1Nuk21.|w wN|21+|u T-1,-1Nu|21.|w wN|21+|u T-1,-1Nu|2B1-1,-1(Id).N22s?|w|Bs-1,-1(Id)+|u|Bs-1,-1(Id)?2.ku uNk1.N(1s)?|w|Bs-1,-1(Id)+|u|Bs-1,-1(Id)?.4QSRUT+VXWZY,#?!_ a b c1d?e fg h i,j k lm(#*?n%o?,i(t)=14i+6Li(t)Li+2(t),P0N=span?0(t),1(t),N2(t)?,Li(t)?i&LegendreZ?,%(#*?XN=?uN(x):uN(x)P0Nd?.sij=Z110j0idt,mij=Z11jidt.pB#D292.1-:440Vol.43sij=(1,i=j,0,i 6=j,mij=12i+3(12i+1+12i+5),j=i,1(2i+7)(2i+3)12i+5,j=i+2,0,#q.?rts?(2.5)(2.6)o$u.v.Id=2J,?(2.5),!?w%xwN=PN2i,j=0wiji(x1)j(x2).W=w00w01w0(N2)w10w11w1(N2).w(N2)0w(N2)1w(N2)(N2),WBW_yl.G#z.(N 1)2_yA$,OvN=l(x1)k(x2),(l,k=0,1,N 2),!:ZDwNvNdx=N2Xi,j=0wijZ11Z11(i(x1)j(x2)(l(x1)k(x2)dx=N2Xi,j=0wijZ11Z11(0i(x1)j(x2),i(x1)0j(x2)(0l(x1)k(x2),l(x1)0k(x2)dx=N2Xi,j=0wijZ110i(x1)0l(x1)dx1Z11j(x2)k(x2)dx2+Z11i(x1)l(x1)dx1Z110j(x2)0k(x2)dx2=N2Xi,j=0wij(slimkj+mliskj)=S(l,:)WM(k,:)T+M(l,:)WS(k,:)T=M(k,:)S(l,:)+S(k,:)M(l,:)W,S(l,:),M(k,:)!?uKvS=sij,M=mijC;lk|,?uKv%$_,SM=(sijM)N2i,j=0;?(2.6),!?w%xuN=PN2i,j=0uiji(x1)j(x2).U=u00u01u0(N2)u10u11u1(N2).u(N2)0u(N2)1u(N2)(N2),UPUtylG?z(N 1)2tyA_$,OhN=l(x1)k(x2),(l,k=0,1,.,N 2),#a,!_ZDuNhNdx=M(k,:)S(l,:)+S(k,:)M(l,:)U,No.5aca:caajc?nm?Galerkin?441?(2.5)o%$CuKv?!(M S+S M)W=F,F=(f00,f10,.,fN2,0;f01,f11,.,fN2,1;.;f0,N2,f1,N2,.,fN2,N2)T,flk=ZDfl(x1)k(x2)dx;(2.6)o%$CuKv?!(M S+S M)U=F,F=(f00,f10,.,fN2,0;f01,f11,.,fN2,1;.;f0,N2,f1,N2,.,fN2,N2)T,flk=ZDwNl(x1)k(x2)dx.Id=3J,?(2.5),!?w%xwN=PN2i,j,q=0wqiji(x1)j(x2)q(x3).Wq=wq00wq01wq0(N2)wq10wq11wq1(N2).wq(N2)0wq(N2)1wq(N2)(N2)WqWqyl#G8z#(N 1)2yA$,W=(W0,W1,WN),Wu)vW8yl Gz(N 1)38yAM$.OvN=l(x1)k(x2)n(x3),(l,k,n=0,1,N 2),!:ZDwNvNdx=N2Xi,j,q=0wqijZD(i(x1)j(x2)q(x3)(l(x1)k(x2)n(x3)dx=N2Xi,j,q=0wqij(slimkjmnq+mliskjmnq+mlimkjsnq)=N2Xm=0S(l,:)WqmnqM(k,:)T+M(l,:)WqmnqS(k,:)T+M(l,:)WqsnqM(k,:)T=N2Xm=0M(k,:)S(l,:)Wqmnq+S(k,:)M(l,:)Wqmnq+M(k,:)M(l,:)Wqsnq=(M(k,:)M(n,:)S(l,:)+S(k,:)M(n,:)M(l,:)+M(k,:)S(n,:)M(l,:)W;?(2.6),!?w%xuN=PN2i,j,q=0uqiji(x1)j(x2)q(x3).Uq=uq00uq01uq0(N2)uq10uq11uq1(N2).uq(N2)0uq(N2)1uq(N2)(N2)442Vol.43Uq0Wq%yl?Gz?(N 1)2%yA$,U=(U0,U1,UN),U0u,vUyltG%zt(N 1)3yA$.OhN=l(x1)k(x2)n(x3),(l,k,n=0,1,N 2),#a,_:ZDuNhNdxdy=(M(k,:)M(n,:)S(l,:)+S(k,:)M(n,:)M(l,:)+M(k,:)S(n,:)M(l,:)U,?(2.5)o%$CuKv?!(M M S+SM M+M SM)W=F,F=(f000,f010,.,f0N2,N2;f(N2)0,0,f(N2)1,0,f(N2)N2,N2)T,fnlk=ZDfl(x1)k(x2)n(x3)dx;(2.6)o%$CuKv?!(M M S+S M M+M S M)U=F,F=(f000,f010,.,f0N2,N2;f(N2)0,0,f(N2)1,0,f(N2)N2,N2)T,fnlk=ZDwNl(x1)k(x2)n(x3)dx.5ZY.1d_e,!_|y f%.!MATLAB R2016b._|iN1.1O u(x)=sinx1sinx2,w(x)=u(x)=22sinx1sinx2,u%#%,w(x),u(x)i(2.1)(2.2)!_f(x).!12y)w(x),u(x)(#*wN(x),uN(x)#.12,p?_,IN 20J,#w(x),u(x)%(_*wN(x),uN(x)_#8 _?M#1013.!1d8,!12+34%INO+J#(?*?,+1234,!1d:_B.,?+,!3y)o.8Md#_.23_#,.81%N,w(x)M8wN(x)%88%C8.N1020304050kw(x)wN(x)k3.7946e-062.0313e-141.6207e-141.2577e-141.4626e-14|w(x)wN(x)|10.00658.7708e-132.5507e-132.8220e-139.2592e-13kw(x)wN(x)k10.00658.7732e-132.5559e-132.8248e-139.2603e-13No.5aca:caajc?nm?Galerkin?44382%N,u(x)M8uN(x)%88%C8.N1020304050ku(x)uN(x)k1.9224e-079.9225e-168.1785e-166.0700e-165.6905e-16|u(x)uN(x)|13.2926e-044.4214e-141.2207e-141.4036e-144.6598e-14ku(x)uN(x)k13.2926e-044.4225e-141.2235e-141.4049e-144.6601e-1483%N,u(x)M8uN(x)%88%C8.N1020304050ku(x)uN(x)k1.2993e-051.0691e-157.6531e-164.0410e-163.7663e-16|u(x)uN(x)|12.8910e-044.3849e-143.3768e-152.8921e-152.2968e-15ku(x)uN(x)k12.8939e-044.3862e-143.4624e-152.9202e-152.3275e-151:N=10C8w(x,y)8wN(x,y)_8.2N=50C8w(x,y)8wN(x,y)_8.3:N=10C8w(x,y)8wN(x,y)_8.4N=50C8w(x,y)8wN(x,y)_8.2O,u(x)=sinx1sinx2sinx3,w(x)=32sinx1sinx2sinx3,u(x)%#%,w(x),u(x)i(2.1)(2.2)!_f(x).!45y)w(x),u(x)(#*wN(x),uN(x)#t.45,p?_,IN 20J,#w(x),u(x)%(_*wN(x),uN(x)_#?M#1012.?!1d_:?.84%N,w(x)M8wN(x)%88%C8.N1015202530kw(x)wN(x)k6.6423e-062.4302e-103.3909e-142.4975e-142.1268e-14|w(x)wN(x)|10.01347.5702e-081.7784e-121.2032e-124.6723e-13kw(x)wN(x)k10.01347.5702e-081.7787e-121.2035e-124.6771e-13444Vol.4385%N,u(x)M8uN(x)%88%C8.N1015202530ku(x)uN(x)k2.2433e-078.2078e-129.1652e-168.8147e-167.3406e-16|u(x)uN(x)|14.5239e-042.5567e-096.0363e-143.9496e-141.6632e-14ku(x)uN(x)k14.5239e-042.5567e-096.0370e-143.9505e-141.6648e-146XXXXB M!%)8.o)?eGalerkin(#*.jk,?=#8?.,H+5 6?%!,(*?q L N.&,lm+(#*?n Io,rs)o%$.u)v?.,!?1?,_1d_e?.B 1d,?d#6?+.N1.1 H ahner P.On the uniqueness of the shape of a penetrable,anisotropic obstacleJ.Journal of Com-putational and Applied Mathematics,2000,116(1):167180.2 Cakoni F,Haddar H.Transmission eigenvalues in inverse scattering theoryJ.2012,hal-00741615(1):527578.3 Colton D,Monk P,Sun J.Analytical and computational methods for transmission eigenvaluesJ.Inverse Problems,2010,26(4):045011.4 Ji X,Xi Y,Xie H.Nonconforming finite element method for the transmission eigenvalue problemJ.Advances in Applied Mathematics and Mechanics,2017,9(1):92103.5 Colton D,P aiv arinta L,Sylvester J.The interior transmission problemJ.Inverse Problems&Imaging,2007,1(1):13.6 Cakoni F,Colton D.Qualitative methods in inverse scattering theory:An introductionM.Berlin:Springer,2005.7 Kirsch A.On the existence of transmission eigenvaluesJ.Inverse Problems&Imaging,2009,3(2):155.8 Yang Y,Bi H,Li H,et al.Mixed methods for the Helmholtz transmission eigenvaluesJ.SiamJournal on Scientific Computing,2016,38(3):A1383A1403.9 An J,Shen J.A Spectral-element method for transmission eigenvalue problemsJ.Journal of Sci-entific Computing,2013,57(3):670688.10 Sun J.Estimation of transmission eigenvalues and the index of refraction from Cauchy dataJ.Inverse Problems,2010,27(1):015009.11 Ji X,Sun J,Turner T.Algorithm 922:a mixed finite element method for Helmholtz transmissioneigenvaluesJ.Acm Transactions on Mathematical 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interior transmission problem and inverse scattering from inhomo-geneous mediaJ.Siam Journal on Mathematical Analysis,1991,22(6):17551762.19 Tan T,Li L,An J.A novel spectral method and error analysis for fourth-order equations in aspherical regionJ.Mathematics and Computers in Simulation,2022,200(c):148161.20 Jiang J,An J,Zhou J.A novel numerical method based on a high order polynomial approximation ofthe fourth order Steklov equation and its eigenvalue problemsJ.Discrete and Continuous DynamicalSystems-B,2022,28(1):5069.21 An J,Li H,Zhang Z.Spectral-Galerkin approximation and optimal error estimate for biharmoniceigenvalue problems in circular/spherical/elliptical domainsJ.Numerical Algorithms,2020,84(2):427455.22 Tan T,Cao W,An J.Spectral approximation based on a mixed scheme and its error estimates fortransmission eigenvalue problemsJ.Computers&Mathematics with Applications,2022,111(1):2033.23 Ren S,Tan T,An J.An efficient spectral-Galerkin approximation based on dimension 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Computing,1994,15(6):14891505.446Vol.43AN EFFICIENT SPECTRAL GALERKIN APPROXIMATIONBASED ON A REDUCED-ORDER FORMAT FORFOURTH-ORDER PROBLEM WITH SIMPLY SUPPORTED PLATEBOUNDARY CONDITIONSQIN Jia-qi,AN Jing(School of Mathematical Sciences,Guizhou Normal University,Guizhou550025,China)Abstract:In this paper,we study an efficient spectral-Galerkin approximation forfourth-order equation with simply supported plate boundary conditions.By introducing anauxiliary function and some appropriate Sobolev spaces,reducing the fourth-order problem totwo coupled second-order problems,establishing the associated weak form and discrete scheme,using Lax-Milgram theorem and the approximation properties of projection operator,we provethe existence and uniqueness of the weak solutions and approximation solutions and the errorestimation between them.Next,by using the orthogonality of Legendre polynomials,we constructa set of appropriate basis functions and derive the matrix formulations based on the tensor-product.Finally,some numerical experiments are carried out to validate the efficiency of the algorithm andthe correctness of the theoretical results.Keywords:fourth-order problem;boundary conditions of simple support plate;reducedformat;spectral method;error estimation2010 MR Subject Classification:65N25;65N30