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GalerkinGalerkinXXX(,710049)(,102206)GalerkinGalerkin,,Euier,,,Galerkin,GalerkinGalerkinO242121NonlinearGalerkinAlgorithmandGalerkinAlgorithmHeYinnian(TheInstituteforScientificComputing,Xi’anJiaotongUniversity,Xi’an710061)YangXiaozhong(DepartmentofFundmentalSciences,North-ChinaElectricPowerUniversity,Beijing102206)AbstractInthispaper,weprovideGalerkinalgorithmandnonlinearGalerkinalgorithmforsolvingthenonlinearevolutionequations,wherethespatialvariablediscretizationisperformedbyGalerkinspectralelementmethodandnonlinearGalerkinspectralelementmethodandthetimevariablediscretizationismadebyEuierexplicitscheme1Moreover,weanalysetheboundedness,stabilityandconvergenceaccuracyofthesealgorithms1Bycomparison,wecometothecondusionthatunderthesameconvergenceaccuracy,thecomputationaltimeandstabilityofnonlinnearGalerkinalgorithmarepriortotheonesofGalerkinalgorithm1KeywordsNonlinearGalerkinAlgorithmGalerkinAlgorithmConvergenceaccuracyBoundednessStability,,,,XXX:1998-05-12199812Dec1199834JOURNALOFXIANINSTITUTEOFPOSTSANDTELECOMMUNICATIONSVol13No14t,,tGalerkin,Galerkin,Galerkin,1HHilbert(,)||AHHD(A)A,A-1HsR,AAs,D(As)|As|,D(As)HilbertV=D(A12),((,))=(A12,A12)=|A12|V,VV,:f-1=supvV(f,v)PfVdudt+Au+B(u,u)=f(111)u(0)=u0(112)0,B(,):VVV,fVA-1,,H{wi},WAwi=1wi012,(i)a(u,v)=(Au,v)=((u,v))PuVb(u,v,w)=(B(u,v),w)Pu,v,wV(m,n),Hm=span{w1,,wm},Hm+n=span{w1,,wm,,wm+n},Hmn=Hm+n/Hn2GalerkinGalerkinPmHVHm,Galerkinum+n(t)Hm+nt0,(ddtum+n,v0)+a(um+n,v0)+b(um+n,um+n,v0)=(f,v0)PvH(211)um+n(0)=Pm+nu0(212)um+nv02199812v0=v+w,vHm,wHmn,um+n=y+z,yHm,zHmn(211),z,ww,w,Galerkin,(y(t),z(t))HmHmn,t0,(dydt,v)+(dzdt,w)+a(y+z,v+w)+b(y,y,v+w)+b(y,z,v)+b(z,y,v)=(f,v+w)PvHm,wHmn(213)y(0)=Pmu0z(0)=Pm+nu0-Pmu0(214)(211)-(212)(213)-(214),Euler,GalerkinGalerkin:Galerkin(G)u0=um+n(0)(215)1t(uk+1-uk,v)+a(uk+1-uk,v)+b(uk,uk,v)=(fk+1,v)PvHm+n(216)Galerkin(NG)y0=y(0)z0=z(0)(217)1t(yk+1-yk,v)+1t(zk+1-zk,w)+a(yk+1+zk+1,v+w)+b(yk,yk,v+w)+b(yk,zk+1,v)+b(zk+1,yk,v)=(fk+1,v+w)PvHn,wHmn(218)t,tk=kt,fk+1=1ttk+1tkf(t)dt,k03u0H,fL(R+;V),GNG311t:t-1m+1,G,t-1mNG(311)GNG,J0|PJ|2M2=2|u0|2+4-11-2f2(312)Jk=1pk2t2|u0|2+2tJ0f(t)2dtf=supt0f(t)-1,,pk=uk(G)yk+zk(NG)E0,k,k0,Pm+nu0fk,Pm+nu0+E0,fk+k,GNGuk+Ek,(yk+ek,zk+k),Ek,(ek,k),E0,k312t:tmin{-1m+n,-2J-1m+n}G,tmin{-1m+n,-2J-1m}NG(313)GNG,J0|EJ|2+Jk=0Ek2t2J(314)34:GalerkinGalerkin|eJ+J|2+Jk=0ek+k2t2J(315),0,10,dkyk,zkuk,2J=exp(2Jk=0dkt){4Jk=0k2t+1|E0|2}4GNGuk,(yk,zk),k0,u(t):U(t)=uk+t-tkt(uk+1-uk)Pt[tk,tk+1]u(t)=yk+zk++t-tkt(yk+1+zk+1-yk-zk)Pt[tk,tk+1]411u0D(A),fL(R+;V),ftL(R+;V)tt-1m+n(411)supt[0,tJ]|u(t)-U(t)|2+tJ0u-U2dtC(tJ)(-2m+n+1+t2)(412)sup|u(t)-u(t)|2+tJ0u-u2dtC(tJ)(-2m+n+1+-3m+1+t2)(413)C(tJ)tJ5411,mn-3m+1-2m+n+1(511)GalerkinGalerkin,311312GalerkinGalerkin,NGGt-1mt-1m+n(512)Jt-2J-1mt-2J-1m+n(513)NGt=O(-1m),,Gt=O(-1m+n),,Gb(uk,uk,v)(m+n)2b(uj,vj,v),NGb(yk,yk,v+w)+b(yk,zk+1,v)+b(zk+1,yk,v)m2+2mnb(ui,vj,v),(9)41998124,CDMAFDMA/CDMAFDMA/CDMA,FDMA/CDMA1T1EngandL1B1Milstein,ComparisonofhybridFDMA/CDMASystemsinfrequencySelectiveRayleighfadingIEEEJ1Select1Commun,1994,12(5):9389512T1EngandL1B1Milstein,CoherentDS/CDMAperformanceNakagamimultipathfadingIEEE,Trans,Commun,1995,43(2,3,4):1143(4)I=NGG=m2+2mn(m+n)2=1-nm+nt+12(514)(511),nm,Galerkin1M1MarionandR1Temam,NonlinearGalerkinMethods,SIAMJ1Numer1Anal,1989,26(5):113911572J1G1HeywoodandT1Rannacher,FiniteElementApproximationoftheNonstationaryNavier-StokesProblem,PartIV,ErrorAnalysisforSecond-OrderTimeDiscretization,SIAMJ1Numer1Anal1,1990,27(2):3533843R1Temam,StabilityAnalysisoftheNonlinearGalerkinMcthod,Math1Comp11991,57(196):47750594:FDMA/CDMA