1、Vol.43(2023)No.5?J.of Math.(PRC)?!#$%&Galerkin()+*-,./(021436527682968;:=,021?0550025)A?B:CD6EGFIHKJ4LMNPO6Q6RPSUT;VUWYXU2V2PG_;4abPcG_edGalerkinfg.hiGjlk2mn4oGpK94q6rGswUWxX6yz46m|4U_KVGWxX,466,Lax-Milgram44q44U_lf6g4,466Hl46q6f6g6U_l6P2U626Iu4_e224.26Legendre;6P_e42262UHl22rIs6_YZIpe9,6IH;4;Z66;4G
2、_YPK6.4,24U4Hl249262,92224UHY2G_Ybc22q62I_e;6.:TeVIWX,J;L4M4NO2Q2R,KVK,d62,224MR(2010)2,CahnHilliard?14,?59?.?UP,?!1013,?!?#?$%?!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?$b6P8Oc?d_P?ef?!,Y?Z227?Ag?68?O?_P?aG
3、alerkine?f.Q?$?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?$I7-?J?!J;Im?.?),v?|?JU/?!g?.Y;|/?7?G+m?maGalerkini?2022-07-18i:2022-09-07?:R.S,H7g?mSobolevm,?RGJm,Lax-Milgram,ef+ypm.t?,yU?SobolevJ,?S?ef?,gef+?.,Legendre?+?7,J?%42u=f(x),x D,u=(x),x D,u
4、=(x),x D,(1.1)S&x Rd(d=2,3),DRd&J68,D(?)68Dg|.Y*+,-:p.2/0G?1Rm#S2.p?.3/,?BWe?f+?+.p?.4/,?3!4?5?6!Ga?7!8?.p?.5/?97?$!.p?.6/?97?A?$?+:?;.2=?BADCEGFIHJIKLMNPOQJmJHm/J2,/?J?SR1.1Tmmm?R,U/Sobolev#S?e?f,?R?.Vw(x)=u(x).W?(1.1)XZYRP:(w(x)=f(x),x D,w(x)=(x),x D,(2.1)(u(x)=w(x),x D,u(x)=(x),x D.(2.2)g?HISobol
5、ev:L2(D):=p:ZDp2dx ;H10(D)=p:p L2(D),xip L2(D),p|D=0,S?!:(p,q)=ZDpqdx,kpk=p(p,p);(p,q)1:=ZDpq+p qdx,kpk1=p(p,p)1.No.5_acbad:ecfagahjickalnmporqtsvuawjxpyzq|npraanprGalerkina435o?N,Y?#?$(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
6、.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!?
7、|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(
8、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
9、,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!#%$?JacobiZ?:Jl,kn(x)=dYi=1Jli,kini(xi),l,k(x)=dYi=1li,ki(xi),n=(n1,n2,.,nd),l=(l1,l2,.,ld),k=(k1,k2,.,kd).!d!N&?Z?#:Q-1,-1N:=spanJ-1,-1n(x):|n|N,|n|=max1idni.d!Ja
10、cobi0)(+*,XN.-:XN=Q-1,-1N.)/)0+12:T-1,-1N:L2-1,-1(Id)Q-1,-1N,3ZId(T-1,-1Nw w)vN-1,-1dx=0,vN Q-1,-1N.(3.4)!d!#4Sobolev,Bsl,k(Id):=w:mxw L2l+s,k+s(Id),0|m|1 s,5 6879!9!:|w|Bsl,k(Id):=(dXi=1ksxiwk2l+sei,k+sei,Id)12,kwkBsl,k(Id):=(X0|m|1skmxwk2l+m,k+m,Id)12,ei:Rd#,A$,|m|1=dPi=1mi,mxw=m1x1m2x2.mdxdw.CB?D
11、288.1E 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(
12、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|B
13、s-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
14、=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 w
15、N|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=Z1
16、1jidt.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)
17、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%$_,SM=(s
18、ijM)N2i,j=0;?(2.6),!?w%xuN=PN2i,j=0uiji(x1)j(x2).U=u00u01u0(N2)u10u11u1(N2).u(N2)0u(N2)1u(N2)(N2),UPUtylG?z(N 1)2tyA_$,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%$CuKv?!(M S+S M)W=F,F=(f00,f10,.,fN2,0;f01,f11,.,fN2,1;.;f0,N2,f1,N2,.,f
19、N2,N2)T,flk=ZDfl(x1)k(x2)dx;(2.6)o%$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$,W=(W0,W1,WN),Wu)vW8yl Gz(N 1)38yAM
20、$.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,:
21、)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,
22、:)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
23、,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
24、)|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
25、(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)_8.2
26、N=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-062.4
27、302e-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
28、.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 a
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30、ng 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
31、 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 S
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33、 method for Helmholtz transmissioneigenvaluesJ.Acm Transactions on Mathematical Software(TOMS),2012,38(4):18.12 An J,Shen J.Spectral approximation to a transmission eigenvalue problem and its applications toan inverse problemJ.Computers&Mathematics with Applications,2015,69(10):11321143.No.5aca:caaj
34、c?nm?Galerkin?44513 Haddar H.The interior transmission problem for anisotropic Maxwells equations and its appli-cations to the inverse problemJ.Mathematical Methods in the Applied Sciences,2004,27(18):21112129.14 Cakoni F,Colton D,Monk P,et al.The inverse electromagnetic scattering problem for aniso
35、tropicmediaJ.Inverse Problems,2010,26(7):074004.15 Cakoni F,Gintides D,Haddar H.The existence of an infinite discrete set of transmission eigenval-uesJ.Siam Journal on Mathematical Analysis,2010,42(1):237255.16 Cakoni F,Haddar H.On the existence of transmission eigenvalues in an inhomogeneous medium
36、J.Applicable Analysis,2009,88(4):475493.17 P aiv arinta L,Sylvester J.Transmission eigenvaluesJ.Siam Journal on Mathematical Analysis,2008,40(2):738753.18 Rynne B P,Sleeman B D.The interior transmission problem and inverse scattering from inhomo-geneous mediaJ.Siam Journal on Mathematical Analysis,1
37、991,22(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 ord
38、er 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):4274
39、55.22 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 reductionscheme for transmis
40、sion eigenvalues in polar geometriesJ.Computers&Mathematics with Appli-cations,2020,80(5):940955.24 Shan W,Li H.The triangular spectral element method for Stokes eigenvaluesJ.Mathematics ofComputation,2017,86(308):25792611.25 Li H,Shan W,Zhang Z.C1-Conforming quadrilateral spectral element method fo
41、r fourth-orderequationsJ.Communications on Applied Mathematics and Computation,2019,1(3):403434.26 An J,Bi H,Luo Z.A highly efficient spectral-Galerkin method based on tensor product for fourth-order Steklov equation with boundary eigenvalueJ.Journal of Inequalities and Applications,2016,2016(1):112
42、.27 Tang T.Spectral and high-order methods with applicationsM.Beijing:Science Press,2006.28 Shen J,Tang T,Wang L L.Spectral methods:algorithms,analysis and applicationsM.New York:Springer,2011.29 Shen J.Efficient spectral-Galerkin method I.Direct solvers of second-and fourth-order equationsusing Leg
43、endre polynomialsJ.Siam Journal on Scientific 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 Uni
44、versity,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 se
45、cond-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 th
46、e 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
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