偏微分方程数值解课件1

–¯ÌIKJJIIJI11�48ˆ£fi¶w«'4òÑ&E†OŽ‰Æ;’‘§?????????????????? ‡©§êŁ)NUMERICALSOLUTIONSTOPARTIALDIFFERENTIALEQUATIONSܪjêƆÚOÆ�SCHOOLOFMATHEMATICSANDSTATISTICScjzhang@mail.hust.edu.cn–¯ÌIKJJIIJI12�48ˆ£fi¶w«'4òÑ8¹1˜Ù>Ł¯KffC©/ª1�Ùý�.Ú�Ô.§ffk�{1nÙý�.§ffk�©{1oÙ�Ô.§ffk�©{1ÊÙV­.§ffk�©{~ë©z–¯ÌIKJJIIJI13�48ˆ£fi¶w«'4òÑ1˜Ù>Ł¯KffC©/ªêÆÔn¥ffC©�nkX4େffnØÚ¢S¿Â§´�E‡©§êŁ){ffÄ:ƒ˜"Buò5n)˜„/ªffC©�n§fl!Äk0�{üffC©¯K§=�g¼êff4Ł¯K"§1�g¼êff4ŁÚ\˜mRnff�g¼êJ(x)=12(A,x)−(b,x),(1)ùpA=(aij)∈Rn×n,x=(x1,x2,···,xn)T,b=(b1,b2,···,bn)T∈Rn,(·,·)deª(½ff˜mRn¥ff˜‡SÈ:(x,y)=n∑i=1xiyi,∀x=(x1,x2,···,xn)T,y=(y1,y2,···,yn)T.–¯ÌIKJJIIJI14�48ˆ£fi¶w«'4òѽ½½nnn1.1eAé¡�½,Ke�ü‡¯K�d:(1)¦x0∈Rn,¦�J(x0)=minx∈RnJ(x)(2)(2)¦e�‚5§|ff)x0∈Rn:Ax=b(3)yyy²²²(1)⇒(2):duJ(x)3fi˜mRn¥��ŁJ(x0),Kx0´J(x)ff4�Ł:,d4Ł3ff7‡^‡k∂J(x0)∂xj=0,j=1,2,...,n,x0=(x(0)1,···,x(0)n)T.=12n∑i=1(aij+aji)x(0)i−bj=0,j=1,2,...,n. Aé¡,Kþ§|=Ax0=b§�x0´§|(3)ff).–¯ÌIKJJIIJI15�48ˆ£fi¶w«'4òÑ(2)⇒(1)µµµex0§|(3)ff),8Ú\˜‡�g¼ϕ(λ)=J(x0+λx),∀x∈Rn,λ∈R.´yϕ(λ)uλ=0���Ł�duJ(x)ux=x0?���Ł§…ϕ(λ)Œ?˜Ú�ϕ(λ)=J(x0)+λ(Ax0−b,x)+λ22(Ax,x).Ïd§dAx0=b9Aé¡�½kϕ(λ)=J(x0)+λ22(Ax,x)=ϕ(0)+λ22(Ax,x)≥ϕ(0),∀λ∈R.�ϕ(λ)uλ=0?���Ł,=J(x)ux0?���Ł.�½n1.1L²:�g¼ffC©¯K†‚5§|ff½)¯KŒ±ƒp=z.–¯ÌIKJJIIJI16�48ˆ£fi¶w«'4òѧ2ü:>Ł¯K§2.1uff²ï�klffu,Ùüàff½3A(0,0),B(l,0)�:,�krݏf(x)ff Ö1R†•eŁ^uuž,uu)C/,âåff²ï�n,ٲu=u(x)Œdeã�:>Ł¯Kx:−Tu′′=f(x),0<x<l,(4)u(0)=u(l)=0.(5)Ù¥T´uffÜå.ã1.1.u�Ė¯ÌIKJJIIJI17�48ˆ£fi¶w«'4òÑ,˜¡,dåÆff4� U�nŒ,uff²ï ˜u∗=u∗(x)´÷v>Ł^‡(5)ff˜ƒŒU ˜¥¦ U��ö.ePuff,˜ ˜u=u(x),KdžÙo UJ(u)=12∫l0(Tu′2−2uf)dx(S! UƒÚ),(6)ÏdkJ(u∗)=minuJ(u).(7)–d,(½uff²ï ˜,�)�«ØÓ/ªffêƯK:1©�:>Ł¯K(4)-(5);2©C©¯K(6)-(7).�öƒm7,3X,«�d'X,ùò´flÙ?˜Ú‡?Øff¯K.°(£ãC{©¯K(7),Äk·‚I‡(½uáuÛ«¼ê˜m.–¯ÌIKJJIIJI18�48ˆ£fi¶w«'4òѧ2.2Sobolev˜m�I=(a,b),I¯=[a,b],L2(I)L«3Iþ²ŒÈffŒß¼ê|¤ff˜m.�L2˜m¥½ÂSÈډê(f,g)=∫baf(x)g(x)dx,f,g∈L2(I),‖f‖=√(f,f)=√∫ba|f|2(x)dx,f∈L2(I),KL2(I)˜‡Hilbert˜m.L2(I)˜˜˜mmmffffffÄÄÄflflfl555ŸŸŸ:�f,g∈L2(I),Kfg∈L1(I)…÷vSchwarzØ�ª|(f,g)|≤‖f‖‖g‖.(8)–¯ÌIKJJIIJI19�48ˆ£fi¶w«'4òÑÒ(6) ó,‡¦J(u)k¿Â,K‡u′,u,f∈L2(I),�¢S¯K=uù«¼êa´Øff,I‡?˜Ú*¿þã¼êa,dÚ\2Â�êVg.^C∞0(I)L«uIþágŒ‡,…3à:a,bff,�S�u"ff¼êa,K∀f∈C1([a,b])9∀ϕ∈C∞0(I),d©ÜÈ©{k∫baf′(x)ϕ(x)dx=−∫baf(x)ϕ′(x)dx,âdÚ\2Â�ê:�f∈L2(I),e3g∈L2(I)¦�∫bag(x)ϕ(x)dx=−∫baf(x)ϕ′(x)dx,∀ϕ∈C∞0(I),(9)K¡g(x)f(x)3Iþff222ÂÂÂ���êêê,Pf′(x)½df(x)dx.555:ef(x)3Ï~¿ÂekáuL2(I)ff�êf′(x),Kf′(x)´f(x)ff2Â�ê,�‡ƒØ,(„P6).Ә‡¼êf(x)ff2Â�ê¿Ø˜,�ÙA�??ƒ�.–¯ÌIKJJIIJI110�48ˆ£fi¶w«'4òÑÚÚÚnnn1.1(CCC©©©{{{ÄÄÄflflflÚÚÚnnn)�g∈L2(I),…∫bag(x)ϕ(x)dx=0,∀ϕ∈C∞0(I),Kg(x)A�??0.yyy²²²{üå„,=Òg∈C(I)ffœ/ƒ±y².3Tœ/e,Œy²g(x)≡0(∀x∈I).eØ,,∃x0∈[a,b]¦�g(x0)6=0,dg(x)3IþffëY5,3x0ff�U(x0,η)⊂(a,b)¦�g(x)3U(x0,η)Sð؏"…�Ò,ؔ�g(x)>0(∀x∈U(x0,η)),�ϕ(x)=exp[−1η2−(x−x0)2],x∈U(x0,η),0,Ù§,Kϕ∈C∞0(I),…∫bag(x)ϕ(x)dx=∫U(x0,η)g(x)·exp[−1η2−(x−x0)2]dx>0.d†®gñ,Ïd(ؤá.–¯ÌIKJJIIJI111�48ˆ£fi¶w«'4òÑ|^TÚnŒ:Ә‡¼êf(x)ff2Â�êA�??ƒ�.d ,¿š?Ûf∈L2(I)ÑkáuL2(I)ff2Â�ê.~~~1.1ÄffiF¼êf(x)=0,−1≤x≤0,1,0<x≤1.ÙáuL2((−1,1)).ef(x)k2Â�êg(x),K∀ϕ∈C∞0((−1,1))k∫1−1g(x)ϕ(x)dx=−∫1−1f(x)ϕ′(x)dx=−∫10ϕ′(x)dx=ϕ(0).l dC©{ÄflÚn,e��ª3(−1,1)SA�??¤á:g(x)=δ(x)=1,x=0,0,x6=0. δ(x)/∈L2((−1,1))(„P7),Ïdg(x)/∈L2((−1,1)).–¯ÌIKJJIIJI112�48ˆ£fi¶w«'4òÑ¢S¯K~~‡¦f†f′þáuL2(I),dÚ\˜mH1(I)={f:f,f′∈L2(I)},¿3Ù¥½ÂÙSȆ‰ê(f,g)1=∫ba[f(x)g(x)+f′(x)g′(x)]dx,∀f,g∈H1(I),‖f‖1=√(f,f)1=√∫ba[f2(x)+(f′(x))2]dx,∀f∈H1(I).ŒyH1(I)´˜‡Hilbert˜m,…¡ƒ˜˜˜ffiffiffiSobolev˜˜˜mmm.?˜ÚŒÚ�þãSobolev˜m˜„œ/,=mffiffiffiSobolev˜˜˜mmm:Hm(I)={f:f(i)∈L2(I),i=0,1,···,m},ÙäSÈډê:(f,g)m=m∑i=0∫baf(i)(x)g(i)(x)dx,∀f,g∈Hm(I),–¯ÌIKJJIIJI113�48ˆ£fi¶w«'4òÑ‖f‖m=√(f,f)m=√√√√m∑i=0∫ba|f(i)(x)|2dx.AOm=0ž,H0(I)Ò´L2(I)˜m,…(f,g)0=(f,g),‖f‖0=‖f‖.§2.34� U�n3fl!,·‚ò|^4� U�n&?˜„ý�>Ł¯K:Lu,−ddx(pdudx)+qu=f,x∈(a,b),(10)u(a)=0,u′(b)=0.(11)Ù¥p,q∈C1(I¯),p(x)≥minx∈I¯p(x)=p¯>0,q(x)≥0(∀x∈I¯),f∈H0(I).�E¼J(u)=12(Lu,u)−(f,u).(12)–¯ÌIKJJIIJI114�48ˆ£fi¶w«'4òÑ|^©ÜÈ©{9>.^‡Œ�J(u)=−12∫baddx(pdudx)udx+12∫baqu2dx−∫bafudx=12∫bap(dudx)2dx+12∫baqu2dx−∫bafudx.?˜ÚÚ\¼a(u,v)=∫ba(pdudxdvdx+quv)dx,(13)KJ(u)=12a(u,u)−(f,u).(14)¼a(u,v)3&?>Ł¯K(10)-(11)éAffC©¯KåX­‡Ł^,Ùä±e5Ÿ:555ŸŸŸ1(‚‚‚555555)a(c1u1+c2u2,v)=c1a(u1,v)+c2a(u2,v),a(u,c1v1+c2v2)=c1a(u,v1)+c2a(u,v2),(c1,c2~ê).–¯ÌIKJJIIJI115�48ˆ£fi¶w«'4òÑ555ŸŸŸ2(ééé¡¡¡555)a(u,v)=a(v,u),∀u,v∈H1(I).555ŸŸŸ3(Lu,v)=a(u,v),u,v∈C2(I).T5Ÿd©ÜÈ©{9>.^‡�Ñ:∫ba[−ddx(pdudx)v+quv]dx=∫ba(pdudxdvdx+quv)dx.d5Ÿ2∼3ŒLééé¡¡¡ŽŽŽfff:(Lu,v)=(Lv,u)=(u,Lv)=(v,Lu).(15)555ŸŸŸ4(���½½½555)3~êγ>0¦�a(u,u)≥γ‖u‖21,∀u∈H1E(I),ùpH1E(I)={u:u∈H1(I),u(a)=0}.–¯ÌIKJJIIJI116�48ˆ£fi¶w«'4òÑyyy²²²∀u∈H1E(I)ku(x)=∫xau′(t)dt,Kdd9SchwarzØ�ª�∫bau2(x)dx=∫ba[∫xau′(t)dt]2dx≤∫ba(x−a)∫xa|u′(t)|2dtdx≤∫ba(x−a)dx·∫ba|u′(t)|2dt=(b−a)22∫ba|u′(x)|2dx.l ∫ba|u′|2dx=12[∫ba|u′|2dx+∫ba|u′|2dx]≥1(b−a)2∫bau2(x)dx+12∫ba|u′|2dx≥γ¯‖u‖21,γ¯=min{12,1(b−a)2},�a(u,u)=∫ba[p(u′)2+qu2]dx≥p¯∫ba(u′)2dx≥γ‖u‖21,γ=p¯γ¯.–¯ÌIKJJIIJI117�48ˆ£fi¶w«'4òÑ^Bd5Ÿ3∼4,k(Lu,u)=a(u,u)≥γ‖u‖21.(16)555ŸŸŸ5(ëëëYYY555)|a(u,v)|≤M‖u‖1‖v‖1,M´†u,vÃ'ff~ê,∀u,v∈H1(I)yyy²²²dëY†lÑSchwarzØ�ª�|a(u,v)|=∣∣∣∣∫ba(pu′v′+quv)dx∣∣∣∣≤M(∫ba|u′||v′|dx+∫ba|u||v|dx)(M=maxx∈I¯{p,q})≤M√∫ba|u′|2dx√∫ba|v′|2dx+√∫ba|u|2dx√∫ba|v|2dx≤M‖u‖1‖v‖1.–¯ÌIKJJIIJI118�48ˆ£fi¶w«'4òÑda(u,v)ff5ŸŒ:∀v∈H1E,∀λ∈R,¼ϕ(λ),J(u∗+λv)÷vϕ(λ)=J(u∗)+λ[a(u∗,v)−(f,v)]+λ22a(v,v).(17)7d,·‚kXeC©�n:½½½nnn1.2�f∈C(I),u∗∈C2(I)´>Ł¯K(10)-(11)ff),KJ(u∗)=minu∈H1EJ(u);(18)‡ƒ,eu∗∈C2(I)∩H1E¦�(18)¤á,Ku∗´>Ł¯K(10)-(11)ff).yyy²²²�u∗∈C2∩H1E,v∈H1Ež,d©ÜÈ©{�a(u∗,v)−(f,v)=∫ba[pdu∗dxdvdx+qu∗v−fv]dx=pdu∗dxvba+∫ba[−ddx(pdu∗dx)+qu∗−f]vdx=p(b)u′∗(b)v(b)+∫ba(Lu∗−f)vdx.(19)–¯ÌIKJJIIJI119�48ˆ£fi¶w«'4òÑeu∗´>Ł¯K(10),(11)ff),KLu∗−f=0,u′∗(b)=0,l d(19)ka(u∗,v)−(f,v)=0,∀v∈H1E.?˜Úd(17)95Ÿ4�J(u∗+λv)=J(u∗)+λ22a(v,v)≥J(u∗),∀λ∈R,v∈H1E.l J(u∗)=minu∈H1EJ(u).‡ƒ,eJ(u∗)=minu∈H1EJ(u),K∀v∈H1E,d(17)-(19)kϕ′(0)=a(u∗,v)−(f,v)=∫ba(Lu∗−f)vdx+p(b)u′∗(b)v(b)=0.AO�v∈C∞0(I),K∫ba(Lu∗−f)vdx=0,∀v∈C∞0(I).–¯ÌIKJJIIJI120�48ˆ£fi¶w«'4òÑâC©{ÄflÚn�Lu∗−f≡0,x∈I,Ïdp(b)u′∗(b)v(b)=0,∀v∈H1E.5¿p(b)≥p¯>0,�v(x)=x−a∈H1E,Kkv(b)>0,l u′∗(b)=0,�u∗>Ł¯K(10)-(11)ff).�5551.2du3ÔnÆ¥¼J(u)~L«ƒAÔnþff U,Ïd¡½n1.24� U�n.3T½n¥,·‚‡¦)u∗�gëYŒ‡(¡ƒ²²²;;;)))).�¢S¯KJ±÷vXdrff1w^‡,dC©¯K(18)5w,LÙ)u∗ëY½©ãëYŒ‡=Œ,Ïd·‚¡�ö>Ł¯Kff222ÂÂÂ))).d ,§(10)¡C©¼J(u)ƒ'ffEuler§§§,>Ł^‡u(a)=0¡rrr›››>>>ŁŁŁ^^^‡‡‡½½½öööflflflŸŸŸ>>>ŁŁŁ^^^‡‡‡, >Ł^‡u′(b)=0¡ggg,,,>>>ŁŁŁ^^^‡‡‡.–¯ÌIKJJIIJI121�48ˆ£fi¶w«'4òѧ2.4Jõ�n½½½nnn1.3�u∈C2(I),Ku´>Ł¯K(10)-(11)ff)ff¿‡^‡´µu∈H1E(I)…÷vC©§a(u,v)−(f,v)=0,∀v∈H1E(I).(20)yyy²²²⇒:d(10)k∫ba(Lu−f)vdx=∫ba[−ddx(pdudx)v+quv−fv]dx=0,∀v∈H1E(I).|^©ÜÈ©{,∀v∈H1E(I)k−∫baddx(pdudx)vdx=−(pdudx)v∣∣∣ba+∫bapdudxdvdxdx=∫bapdudxdvdxdx.Ïda(u,v)−(f,v)=∫ba(pdudxdvdx+quv−fv)dx=0,∀v∈H1E(I).–¯ÌIKJJIIJI122�48ˆ£fi¶w«'4òÑ⇐:d(19)ka(u,v)−(f,v)=p(b)u′(b)v(b)+∫ba(Lu−f)vdx,∀v∈H1E(I).â(20)�p(b)u′(b)v(b)+∫ba(Lu−f)vdx=0,∀v∈H1E(I)./Ïuþª9aqu½n1.2ff¿©5ffy²Œ�Lu−f=0,x∈I;u′(b)=0.q®u∈H1E(I),�u´>Ł¯K(10)-(11)ff).�5551.3(20)ff†>3åÆ¥~~L«Jõ,Ïd¡½n1.3JJJõõõ���nnn.�u÷v>Ł¯K(10)-(11)ž,¡ƒ>Ł¯K(10)-(11)ff²²²;;;))),eu÷v(20)ffš1w),K¡ƒ>Ł¯Kff222ÂÂÂ))).Jõ�n' U�nA^2,§Ø=·^ué¡�½Žf§,·^ušé¡�½Žf§.–¯ÌIKJJIIJI123�48ˆ£fi¶w«'4òѽn1.3„Œ±í2�˜„ff§œ/,…a(u,v)ؘ½‡¦é¡�½.½½½nnn1.4�u∈C2(I),p∈C1(I),pmin>0,r,q∈C(I),f∈L2(I),Ku÷v>Ł¯KLu=−ddx(pdudx)+rdudx+qu=f,x∈(a,b),u(a)=0,u′(b)=0,(21)ff¿‡^‡´:u∈C2(I)∩H1E(I)…÷vC©§a(u,v)−(f,v)=0,∀v∈H1E(I),Ù¥a(u,v)=∫ba(pdudxdvdx+rdudxv+quv)dx.–¯ÌIKJJIIJI124�48ˆ£fi¶w«'4òѧ3�ffiý�>Ł¯K§3.1Sobolev˜mHm(G)fl!±�‘«~éþ!0�ffSobolev˜m92Â�ê�VgŁ²1í2.d,ð�Gk.²¡«,Γ´Gff>.,Ù´©ã1wff{ü4­‚,…PG¯=G∪ÏGff4,éuG¯þ?˜¼êu,¡8Üsupp{u(x,y)},{(x,y)∈G¯:u(x,y)6=0}ff4uff|||888.esupp{u(x,y)}⊂G,K¡u3G¥äk;;;———|||888.Œyµäääkkk;;;———|||888ffffff¼¼¼êêê777333Γffffff,,,���SSSððð0.–¯ÌIKJJIIJI125�48ˆ£fi¶w«'4òÑPC∞0(G)GþágŒ‡…k;—|8ff¼êa¶L2(G)3Gþ²ŒÈffŒß¼ê˜m,ÙSÈډê©O½Â(f,g)=∫Gf·gdxdy,‖f‖=√(f,f)=√∫G|f|2dxdy,∀f∈L2(G),e∃gˆ,h∈L2(G)¦�∫Ggˆϕdxdy=−∫Gf∂ϕ∂xdxdy,∫Ghˆϕ=−∫Gf∂ϕ∂ydxdy,∀ϕ∈C∞0(G),K¡fkéx,yff˜˜˜ffiffiffi222   ���êêê,©OP∂f∂x=gˆ,∂f∂y=h.g„Œ½Âpffi2Â�ê.d ,·‚½ÂGþffSobolev˜˜˜mmmH1(G)={f:f,fx,fy∈L2(G)}9ÙSÈډê(f,g)1=∫G(fg+fxgx+fygy)dxdy,‖f‖1=√(f,f)1.aq/,„Œ½ÂpgSobolev˜mHm(G),mšK�ê.–¯ÌIKJJIIJI126�48ˆ£fi¶w«'4òѧ3.24� U�nÄPoisson§ff1˜>Ł¯K−∆u=f(x,y),(x,y)∈Gu|Γ=0,(22)Ù¥∆=∂2∂x2+∂2∂y2LaplaceŽf.Ú\¼J(u)=12(−∆u,u)−(f,u),(23)¿Pdσ=dxdy,∂u∂~nu÷>.Γffü  {•~nff•�ê,K|^Green1˜úª∫G(−∆u)vdσ=∫G(∂u∂x∂v∂x+∂u∂y∂v∂y)dσ−∫Γv∂u∂~nds(24)9(22)¥ff>.^‡Œ�J(u)=12∫G[(∂u∂x)2+(∂u∂y)2]dσ−∫Gfudσ.(25)–¯ÌIKJJIIJI127�48ˆ£fi¶w«'4òÑPa(u,v)=∫G(∂u∂x∂v∂x+∂u∂y∂v∂y)dσ,(26)Kd(25)kJ(u)=12a(u,u)−(f,u).(27)¼a(u,v)3&?>Ł¯K(22)ƒAffC©¯KkX­‡Ł^,Ù䘑œ/¥a(u,v)aqff5Ÿ.555ŸŸŸ1(‚‚‚555555)a(c1u1+c2u2,v)=c1a(u1,v)+c2a(u2,v),a(u,c1v1+c2v2)=c1a(u,v1)+c2a(u,v2).555ŸŸŸ2(ééé¡¡¡555)a(u,v)=a(v,u).555ŸŸŸ3(−∆u,v)=a(u,v),∀v∈H10(G),{v∈H1(G):v|Γ=0}.T5ŸŒd(24),(26)�Ñ.–¯ÌIKJJIIJI128�48ˆ£fi¶w«'4òÑ555ŸŸŸ4(���½½½555)∃γ>0¦�a(u,u)≥γ‖u‖21,∀u∈H10(G).555ŸŸŸ5(ëëëYYY555)|a(u,v)|≤M‖u‖1‖v‖1,M†u,vÃ'ff~ê.éuC©¯K:¦u∗∈H10(G)¦�J(u∗)=minu∈H10(G)J(u).(28)·‚kXeC©�n:½½½nnn1.5�u∗∈C2(G¯)∩H10(G),Ku∗>Ł¯K(22)ff)�…=�(28)¤á.–¯ÌIKJJIIJI129�48ˆ£fi¶w«'4òÑyyy²²²∀u∈H10(G),Ú\¹¢ëêλff¼êϕ(λ)=J(u∗+λu).|^a(u,v)ff5ŸŒ�ϕ(λ)=J(u∗)+λ[a(u∗,u)−(f,u)]+λ22a(u,u).eu∗∈C2(G¯)∩H10(G)´>Ł¯K(22)ff),Kâ5Ÿ3�a(u∗,u)−(f,u)=(−∆u∗−f,u)=0,∀u∈H10(G).lJ(u∗+λu)=J(u∗)+λ22a(u,u)≥J(u∗),∀u∈H10(G),∀λ∈R.�J(u∗)=minu∈H10(G)J(u).Ù¿©5ffy²†½n1.2ffaq.–¯ÌIKJJIIJI130�48ˆ£fi¶w«'4òѽn1.5¡444���   UUU���nnn,T½n‡¦u∗∈C2(G¯),�¡ƒ���;;;))). C©¯K(28)K#NkØáuC2(G¯)ff),¡ƒ222ÂÂÂ))).e(22)¥>.^‡“ƒ±šàg>.^‡u|Γ=ϕ(x,y),ϕ∈C1(Γ),(29)KŒ�A½¼êu0∈C2(G¯),u0|Γ=ϕ,-v=u−u0,Ïd>.^‡(29)�àgz,…v÷v§−∆v=f+∆u0.(30)8Ú\¼Ĵ(v)=12∫G(−∆v)vdxdy−∫G(f+∆u0)vdxdy.–¯ÌIKJJIIJI131�48ˆ£fi¶w«'4òÑâ½n1.5,u∗=v∗+u0e�šàg>Ł¯Kff):{−∆u=f(x,y),(x,y)∈Gu|Γ=ϕ(x,y),ϕ∈C1(Γ)(31)�duĴ(v∗)=minv∈H10(G)Ĵ(v). |^Green1˜úª9>.^‡(29)Œ�Ĵ(v)=J(u)+~ê.�šàg>Ł¯K(31)†C©¯KJ(u∗)=minu∈H1(G)u|Γ=ϕJ(u)(32)�d.e¯K(22)¥ff>.^‡“ƒ±1�½1na>.^‡(∂u∂~n+au)Γ=0,a≥0,(33)–¯ÌIKJJIIJI132�48ˆ£fi¶w«'4òÑKÏLÚ\¼J(u)=12a(u,u)−(f,u),u∈H1(G),Ù¥a(u,v)=∫G(∂u∂x∂v∂x+∂u∂y∂v∂y)dxdy+∫Γauvds,/Ïu½n1.2aqffy²Œ�:½½½nnn1.6>Ł&m