5 The Laplace Equation in a convex polygon

Transcription

1 5 Te Laplace Equaton n a convex polygon Te most mportant ellptc PDEs are te Laplace, te modfed Helmoltz and te Helmoltz equatons. Te Laplace equaton s u xx + u yy =. (5.) Te real and magnary parts of an analytc functon satsfy Laplace s equaton. Indeed, f f(z) s an analytc functon, f(z) =u R (x, y)+u I (x, y), u R,u I real, ten u R and u I satsfy te Caucy-Remann I (5.2) D erentatng te above equatons wt respect to x and y respectvely, and ten addng te resultng expressons we fnd tat u R satsfes te Laplace equaton u Rxx + u Ryy =. (5.3) Smlarly, u Ixx + u Iyy =. (5.4) If u satsfes te Laplace equaton (5.), ten u s called a armonc functon. Tradtonally, armonc functons are assocated wt te real and magnary parts of an analytc functon. However, tere s an alternatve drect way to assocate armonc and analytc functons: te functon u(x, y) (wc may be complex) s armonc u z s analytc. Indeed, f u z s analytc ten u z z =,.e., u s armonc; te nverse s also true. Te Global Relaton Recall tat te frst step of te new metod conssts of rewrtng te gven PDE as a famly of conservaton laws and ten usng Green s dentty to obtan te global relaton. For ellptc PDEs nvolvng second order dervatves, one needs two global relatons. However, f we assume tat u s real, ten te second global relaton can be obtaned from te frst va complex conjugaton. 44

2 In order to derve a global relaton we frst consder te adjont of te Laplace equaton, wc s clearly tself, v xx + v yy =. (5.5) Multplyng equatons (5.) and (5.5) by v and u respectvely, and ten subtractng te resultng equatons we fnd (vu x uv x ) x +(vu y uv y ) y =. Lettng v =exp( x + y), wc s a partcular soluton of (5.5) for any complex constant, we fnd te famly of conservaton laws e x+ y (u x + u) x + e x+ y (u y u) =, 2 C. (5.6) Te exponental exp( x+ y) provdes an oter partcular soluton of (5.5), and ts yelds e x+ y (u x u) x + e x+ y (u y u) =, 2 C. (5.7) y We note tat f u s real, ten equaton (5.7) can be obtaned from (5.6) by takng te complex conjugate and ten replacng n te resultng equaton by. Ts procedure s called Scwartz conjugaton. Suppose tat te Laplace equaton s vald n te doman. Ten, equatons (5.6) and (5.7) togeter wt Green s teorem, mply te followng global relatons: e x+ y [(u x + u)dy (u y u)dx] =, 2 C, (5.8) and e x+ y [(u x u)dy (u y u)dx] =, 2 C, (5.9) were denotes te boundary of. Te most well known boundary value problems for ellptc PDEs are eter te Drclet problem were u s prescrbed on te boundary, or te Neumann problem were te normal dervatve, denoted by u!,sprescrbed on te boundary. In order to rewrte te global relatons n terms of u and u!, we parameterze te boundary n terms of ts arclengt wc we denote by s. Ten, f u T denotes te dervatve of u along te tangent to, and y 45

3 u! denotes te dervatve of u normal to u T n te outward drecton, ten d erentatng u(x(s),y(s)) we fnd u x dx + u y dy = u T ds. (5.) Snce te nfntesmal vector (dy, dx) s normal to te nfntesmal vector (dx, dy), we fnd u x dy u y dx = u! ds. (5.) Tus, we can rewrte equatons (5.8) and (5.9) n terms of u and u! : (u x + u)dy (u y u)dx = u! ds + u(dx + dy). Hence, te global relaton (5.8) becomes apple e x+ y u! + u dx ds + dy ds Smlarly Lettng e x+ y apple u! + equatons (5.2) and (5.3) become e z u! + and u dx ds dy ds ds =. (5.2) ds =. (5.3) z = x + y, z = x y, (5.4) e z u! + Alternatve Global Relatons Recall tat equatons u dz ds u d z = 2 ds =, (5.5) ds @y. (5.7) Hence, te Laplace equaton (5.) can be rewrtten n te form Ts equaton mmedately mples u z z =. (5.8) e z u z z =, 2 C, 46

4 wc states tat te functon e z u z, s an analytc functon. Hence, Caucy s teorem yelds e z u z dz =, 2 C. (5.9) Smlarly, e z u z z =, and ten Green s teorem mples e z u z d z =, 2 C. (5.2) Equatons (5.5) and (5.6) nvolve u and u!, wereas equatons (5.9) and (5.2) nvolve u T and u!. Tus, for a Drclet or a Neumann boundary value problem te former equatons are more convenent, wereas f only dervatves are prescrbed as boundary condtons, ten equatons (5.9) and (5.2) provde a better coce. A Polygonal Doman Let be te nteror of te polygonal doman specfed by te complex numbers z, z 2,...,z n, z n+ = z. Fgure 5. Let L j denote te sde (z j,z j+ ). Ten, te global relaton (5.5) becomes nx Ŵ j + j= nx j= ˆD j =, 2 C, (5.2) 47

5 were {W j } n denote te transforms of te Neumann boundary values and {D j } n denote te transforms of te Drclet boundary values: and Ŵ j = ˆD j = zj+ z j e z u wj ds, j =, 2,...,n, 2 C (5.22) zj+ z j e z dz u j ds, j =, 2,...,n, 2 C. (5.23) ds If u s real, ten nstead of analysng te global relaton (5.6), we can analyse te complex conjugate of equaton (5.2). Tus, for real u, equaton (5.2) and ts complex conjugate provde two equatons for n unknown functons, snce for a well posed problem only one boundary condton s gven on eac sde. Ts stuaton appears omnus, owever n equaton (5.2) te complex constant s arbtrary, tus n ts sense equaton (5.2) contans nfntely many equatons. It turns out tat ts observaton provdes a most e cent way for te numercal ntegraton of ts problem. Integral Representatons Recall tat te second step of te new metod nvolves constructng an ntegral representaton for te soluton, wc nvolves ntegrals defned n te complex -plane. For te Laplace equaton te followng result s vald. Suppose tat s te nteror of a convex polygon wt corners at z,z 2,...,z n, z n+ = z, see fgure 5.. were Ten, and te rays {l j } n u z = 2 û j ( )= nx j= zj+ are defned by l j e z û j ( )d, z 2, (5.24) z j e z u z dz, 2 C, (5.25) l j = { 2 C, < <, arg = arg(z j+ z j )}, j =, 2, 3,...,n, and are orented towards nfnty. We empasze tat usng ts notaton te global relaton (5.9) takes te form nx û j ( )=, 2 C. (5.26) j= 48

6 5. Te Laplace equaton on te quarter plane Let be te frst quadrant of te complex z-plane,.e., < arg z< /2. Fgure 5.2 Ten, te relevant transforms are: û ( )= 2 snce on ts contour z = y and dz = dy. Also, û 2 ( )= 2 snce on ts contour z = x and dz = dx. Note tat arg l = arg(z 2 z )= ( Tus, te ntegral representaton reads u z = e z û ( )d + e y [u x (,y)+u y (,y)] dy, < apple, e x [u x (x, ) u y (x, )] dx, = apple, and te global relaton takes te form 2 )= 2, arg l 2 = arg(z 3 z 2 )=. e z û 2 ( )d, < arg z< 2, (5.27) û ( )+û 2 ( )=, apple arg apple 3 2. Oblque Boundary Condtons Let te real-valued functon u(x, y) satsfy te Laplace equaton n te frst quarter plane, < arg z< /2, see fgure

7 Suppose tat te dervatve of te functon u s prescrbed along te drecton makng an angle wt te sdes of te doman, see fgure 5.3: u x (,y)sn + u y (,y) cos = g (y), <y<, (5.28) u y (x, ) sn + u x (x, ) cos = g 2 (x), <x<. (5.29) Fgure 5.3 We wll sow tat te soluton of te Laplace equaton s gven by u z = e z e [G ( )+G 2 ( )] d + e z e [G ( )+G 2 ( )] d, (5.3) were te functons G and G 2 can be computed n terms of te gven data va te expressons G ( )= 2 e y g (y)dy, G 2 ( )= 2 e x g 2 (x)dx, < apple. (5.3) Proof Let u (y) and u 2 (x) denote te unknown dervatves n te drectons normal to te drectons of te gven dervatves,.e., u y (,y)sn u x (,y) cos = u (y), <y<, (5.32) u x (x, ) sn + u y (x, ) cos = u 2 (x), <x<. (5.33) Solvng equatons (5.28) and (5.32) for {u x (,y), u y (,y)}, as well as equatons (5.29) and (5.33) for {u x (x, ), u y (x, )} we fnd te followng expressons u y (,y)=g (y) cos + u (y)sn, 5

8 u x (,y)=g (y)sn u (y) cos, u y (x, ) = g 2 (y)sn + u 2 (x) cos, u x (x, ) = g 2 (x) cos + u 2 (x)sn. Substtutng tese expressons n te defntons of û ( ) and û 2 ( )wefnd û ( )=e [G ( )+U ( )], û 2 ( )=e [G 2 ( )+U 2 ( )], (5.34) were G and G 2 are te known functons defned n (5.3), wereas U and U 2 denote te transforms of te unknown functons u and u 2,.e., U ( )= e y u (y)dy, U 2 ( )= e x u 2 (x)dx, < apple. 2 2 (5.35) Substtutng te expressons for û and û 2 n te global relaton and also takng te Scwartz conjugate of te resultng equaton (.e. complex conjugaton followed by 7! )wefnd G ( )+U ( )+G 2 ( )+U 2 ( )=, apple arg apple 3 2, (5.36) G ( ) U ( )+G 2 ( ) U 2 ( )=, apple arg apple. (5.37) 2 Tese equatons are two equatons for te tree unknown functons U ( ), U 2 ( ), U 2 ( ). Te representaton for u z nvolves ntegrals along te boundary of te frst quadrant of te complex -plane. In ts doman te functon U 2 ( ) s analytc, tus we wll express û and û 2 n terms of ts functon (we can also coose U ( ) nstead of U 2 ( )snceu ( ) s also analytc n te frst quadrant) equaton (5.37) yelds: U ( )= U 2 ( )+G ( )+G 2 ( ), apple arg apple. (5.38) 2 In order to determne U 2 ( ) we elmnate from equatons (5.36) and (5.37) te functon U ( ) and ten we replace by n te resultng equaton: U 2 ( )=U 2 ( )+2G ( )+G 2 ( )+G 2 ( ), >. (5.39) Substtutng te expressons for U ( ) and for U 2 ( û and û 2,.e., n equatons (5.34), we fnd ) n te formula for û ( )=e [2G ( )+G 2 ( ) U 2 ( )], 2 l, 5

9 Fgure 5.4 û 2 ( )=e [2G 2 ( )+G 2 ( )+2G ( )+U 2 ( )], 2 l 2, (5.4) were te rays l and l 2 are sown n Fgure 5.4. Te terms nvolvng U 2 ( ), wc are sown n Fgure 5.5, yeld a zero contrbuton to u z. Indeed, te real part of exp[ z] equals exp[ Rx Iy], tus ts exponental s bounded n te frst quadrant of te complex -plane. Furtermore, te functon U 2 ( ) s analytc and of order O as!. Tus, Jordan s lemma appled to te frst quadrant of te complex -plane mples tat te contrbuton of ts unknown functon vanses. Fgure 5.5 Usng Jordan s lemma n te frst quadrant of te complex -plane we can transform te contrbuton of te term nvolvng G 2 ( ) from te ntegral along l 2 to te ntegral along l and ence equaton (5.25) becomes te 52

10 expressons appearng n te ntegrals of te rs of equaton (5.3). Example 5. Consder te followng partcular case of te above problem: Ten, g (y) =e a y, g 2 (x) =e a 2x, a, a 2 postve constants. G ( )= 2 Tus, equaton = e 2 apple We note tat e ( a )y dy = 2 e z d a a 2 a, G 2 ( )= 2 e z = e z sn(arg +arg z). a 2. e z + a d + a 2. (5.4) Furtermore, te poston poles of eac ntegrand, wc are depcted n Fgure 5.6, yelds te analytcty n te frst quadrant. Fgure 5.6 Hence, n order to aceve exponental decay as!for all z, we can coose a ray n te -plane suc tat < arg + arg z<, were z satsfes apple arg z apple 2. 53

11 Tus, te ntegral representaton of te soluton takes te = e 2 wc = e e z a a 2 + a + e d, (5.42) + a 2 e z a 2 a 2 + a a 2 d, (5.43) 2 wt 2 (, /2). 5.2 Numercal consderatons Te numercal soluton of te global relatons for determnng te unknown boundary values nvolves te followng two steps.. Expand te functon [u j ] denoted by {S l (t)} N : u n terms of N bass functons u j NX l= a j l S l(t), j X b j l S l(t), l= j =, 2,...,n. A convenent suc bass s gven by te Legendre polynomals of order l, denote by P l. Let Ŝl( ) denote te Fourer transform of S l (t), namely Ŝ l ( )= e t S l (t)dt, 2 C. (5.44) For te Legendre polynomals te relevant Fourer transform can be computed explctly, ˆP l ( )= e t P l (t)dt = lx k= apple (l + k)! ( ) l+k e e (l k)!k! (2 ) k+. (5.45) Ten, te global relaton and ts complex conjugate yeld two equatons nvolvng te constants a j l and b j l. By evaluatng tese equatons at approprately cosen values of called collocaton ponts, we can solve for te unknown coe cents. 54

12 Te case of te Square Consder te Laplace equaton n te nteror of te square wt corners z = +, z 2 =, z 3 =, z 4 =+. Ten, te global relaton (5.2) nvolves te followng terms: û ( )=e e y u () x + u () dy, û 2 ( )=e e x u (2) y + u (2) dx, û 3 ( )=e e y u (3) x + u (3) dy, Let û 4 ( )=e e x u (4) y + u (4) dx. (5.46) Ŵ ( )= e t W (t)dt, ˆD( )= e t D(t)dt, 2 C, (5.47) were W (t) and D(t) denote Neumann and Drclet boundary values respectvely. Ten, û ( )= e Ŵ ( )+ ˆD ( ), û 2 ( )=e Ŵ2 ( )+ ˆD 2 ( ), û 3 ( )=e Ŵ3 ( )+ ˆD3 ( ), û 4 ( )=e Te approxmate global relaton yelds Ŵ4 ( ) ˆD4 ( ). (5.48) û ( )+û 2 ( )+û 3 ( )+û 4 ( )=, 2 C, (5.49) were NX û ( ) e l= a l ˆP l ( )+b l ˆP l ( ), 55

13 û 2 ( ) e NX l= NX û 3 ( ) e û 4 ( ) e NX l= l= a l ˆP 2 l ( )+b l ˆP 2 l ( ), a l ˆP 3 l ( )+b l ˆP 3 l ( ), a l ˆP 4 l ( )+b l ˆP 4 l ( ). (5.5) 2. For a gven sde, coose n suc a way tat for te gven sde we obtan te usual Fourer transform of te Legendre functons, wereas te contrbuton from te remanng sdes vanses as!(t can be sown tat for a convex polygon suc a coce s always possble). In fact we coose to be on te complement of te rays l j.tsensures tat for a gven sde we get te usual Fourer transform for te Legendre functons, wereas te contrbuton from te remanng sdes vanses as!. sde. Multply (5.49) by e and ten let =, >. 4 We fnd te followng forms for Ŵj (and smlarly for ˆDj ): Ŵ ( ), e e Ŵ 2 ( ), e 2 Ŵ 3 ( ), e e Ŵ 4 ( ). Te frst terms nvolve te FT, wereas te remanng terms vans as!. Ts s obvous for te trd term, wereas te second and te fourt terms nvolve te ntegral e (+t) w(t)dt; snce <t<, t follows tat + t>, tus exp[ ( + t)] vanses as!. sde 2. Multply by e and ten let =, >. sde 3. Multply by e and ten let =, >. sde 4. Multply by e and ten let =, >. 56

14 For we can use te dscrete values = R M m, m =, 2,...,M, R>, were R/M determnes ow close are te collocaton ponts. We fnd numercally te followng rules for low condton number: R M 2, M Nn. 57