任意凸四边形区域上二阶变系数椭圆边值问题有效的谱Galerkin逼近.pdf

Advances in Applied Mathematics A?,2024,13(1),414-429Published Online January 2024 in Hans.https:/www.hanspub.org/journal/aamhttps:/doi.org/10.12677/aam.2024.131042?o/?CX?Kk?Galerkin%C444?AAA B“?B B?vF2023c12?25FF2024c1?19FuF2024c1?26F?J?3?o/?CX?K?k?Galerkin%CkLV5?CIC?o/=?D=1,123D?f/9A?lgy?f)?35,?|Legendre?E?%Cm|k?l?/?L?y?Galerkin%C?o/?CX?K?c?o/?CX?Kf)?35Galerkin%CEfficient Spectral Galerkin Approximationfor Second Order Elliptic Boundary ValueProblems with Variable Coefficient onArbitrary Convex Quadrilateral DomainXuelin Liu,Yinghong Zhang,Fang ShiSchool of Matnematical Sciences,Guizhou Normal University,Guiyang Guizhou:4?,A,.?o/?CX?K?Galerkin%CJ.A?,2024,13(1):414-429.DOI:10.12677/aam.2024.1310424?Received:Dec.25th,2023;accepted:Jan.19th,2024;published:Jan.26th,2024AbstractIn this paper,an efficient spectral Galerkin approximation for second-order ellipticboundary value problems with variable coefficients on an arbitrary convex quadrilat-eral region is proposed.Firstly,any quadrilateral region is converted toD=1,12by bilinear isoparametric transformation and coordinate transformation,and its weakform and corresponding discrete format onD are established.Secondly,we prove theexistence and uniqueness of the weak solution.In addition,the Legendre orthogonalpolynomial is used to construct a set of effective basis functions in the approximationspace,and the matrix form of the discrete scheme is derived.Finally,the spectralconvergence of spectral Galerkin approximation to the second-order elliptic boundaryvalue problem with variable coefficients on arbitrary convex quadrilateral region isverified by numerical experiments.KeywordsArbitrary Convex Quadrilateral Region,Second Order Elliptic Boundary ValueProblem with Variable Coefficient,Existence and Uniqueness of Weak Solutions,Spectral Galerkin Approximation,Spectral ConvergenceCopyright c?2024 by author(s)and Hans Publishers Inc.This work is licensed under the Creative Commons Attribution International License(CC BY 4.0).http:/creativecommons.org/licenses/by/4.0/1.3O+I)?5?KA?KXf!6NDA?K 15?5K5z?8(E/)?Ku?K?OkJXk?6,7!k?811 1215?O()v?1wk?:2?$u?O3/)?dT?DOI:10.12677/aam.2024.131042415A?4?O?K?5?5S3z 16 J?3Maxwell DA?K?k?%Cz 17J?o/?A?Kup?%C?,?vkf)?35?)u3?o/?CX?K?Galerkin%C?d?J?J?3?o/?CX?K?k?Galerkin%CkLV5?CIC?o/=?3?f/9A?lgy?f)?35,?|Legendre?E?%Cm|k?lA?/?LMatlab?(JL?Galerki%CU?k?)?o/?CX?K?|Xe312!?K3L?C?f/9l313!y?)?35314!?l?k?y15!?316!?(5?52.C?f/9l?.e?K(u(x,y)+(x,y)u(x,y)=f(x,y),(x,y)u(x,y)=0,(x,y)(1)d:I(x1,y1),(x2,y2),(x3,y3),(x4,y4)?o/(x,y)?Kk.-Hs()Hs0()L?s?Sobolevmkks,|s,OLd?(1)?f/:u H10()?a(u,v)=f(v),v H10(),(2)a(u,v)=Ruv+uvdxdy,f(v)=Rfvdxdy-(x=x1+(x2 x1)x+(x3 x1)y+(x4 x3 x2+x1)x y,y=y1+(y2 y1)x+(y3 y1)y+(y4 y3 y2+y1)x y,(3)w,(3)lD=(x,y):x,y 0,1?V5?CP u(x,y)=u(x,y),(x,y)=(x,y)?y?C?C?Jacobi1?Ive?J(x,y)=?x2 x1+a yx3 x1+a xy2 y1+b yy3 y1+b x?0(4)DOI:10.12677/aam.2024.131042416A?4?a=x4 x3 x2+x1,b=y4 y3 y2+y1.q-x=12(+1),y=12(+1).(5)w,(5)lD=(,):,1,1?D?ICP u(,)=u(x,y),J(,)=J(x,y),(,)=(x,y)d(3)(5)9E?K?FfLCL?d/u=u=2 uy3y1+b2(+1)J uy2y1+b2(+1)J,ux2x1+a2(+1)J ux3x1+a2(+1)J(6)Sobolevm:H10(D)=u:RD(|uy3y1+b2(+1)J uy2y1+b2(+1)J|2+|ux2x1+a2(+1)J ux3x1+a2(+1)J|2)Jdd/?=1,o:IOA(x1,y1)=(2,1),B(x2,y2)=(2,3),C(x3,y3)=(1,1.25),D(x4,y4)=(2,1)X 1OXeK?)(u+u=f,(x,y)u=0,(x,y)TK?()(y+12x+2)(x 2)(y+112x 76)(y 94x 72)sinxsinyu?N3 2x?()u%C)uNm?-e(u,uN)l 2*?N 18%C)()m?1012?*/L?5O3 3 4?()N=20%C”35 6 O?N=15N=20()?)m?”l 1-6*?Figure 1.Generally convex quadrilateral area 1.o/DOI:10.12677/aam.2024.131042420A?4?Figure 2.Error curve between approximate andanalytic solutions 2.%C)m?-Figure 3.Exact solution of u(x,y)image 3.()u(x,y)?”Figure 4.Image of the approximation solution at N=20 4.N=20%C)?”DOI:10.12677/aam.2024.131042421A?4?Figure 5.An image of the error between the exact solutionand the approximate solution at N=15 5.()N=15%C)m?”Figure 6.An image of the error between the exact solutionand the approximate solution at N=20 6.()N=20%C)m?”2?o/?=5,o:IOA(x1,y1)=(0,1),B(x2,y2)=(3,0),C(x3,y3)=(1,0),D(x4,y4)=(0,1)X 7OXeK?)(u+u=f,(x,y)u=0,(x,y)TK?()(x+y+1)(y x 1)(y+13x 1)(y 13x+1)sinxsinyu?N3 8x?()u%C)uNm?-e(u,uN)l 8*?N 16%C)()m?1012?*/L?5O3 9 10?()N=20%C”3DOI:10.12677/aam.2024.131042422A?4?11 12O?N=15N=20()?)m?”l 7-12*?Figure 7.Generally convex quadrilateral area 7.o/Figure 8.Error curve between approximate and analytic solutions 8.%C)m?-DOI:10.12677/aam.2024.131042423A?4?Figure 9.Exact solution of u(x,y)image 9.()u(x,y)?”Figure 10.Image of the approximation solution at N=20 10.N=20%C)?”Figure 11.An image of the error between the exactsolution and the approximate solution at N=15 11.()N=15%C)m?”DOI:10.12677/aam.2024.131042424A?4?Figure 12.An image of the error between the exactsolution and the approximate solution at N=205 12.()N=20%C)m?”3?o/?(x,y)=xy,o:IOA(x1,y1)=(0,1),B(x2,y2)=(3,0),C(x3,y3)=(1,0),D(x4,y4)=(0,1)X 13OXeK?)(u+(x,y)u=f,(x,y)u=0,(x,y)TK?()(x+y+1)(y x 1)(y+13x 1)(y 13x+1)sinxsinyu?N3 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