S. G. Hong / Elecronic Journal of Boundary Elemens, Vol. 3, No., pp. -32 26 Symmeric Complex Boundary Elemen Scheme for 2-D Sokes Mixed Boundary Value Problem Sun-Gwon Hong Insiue of Mahemaics, Academy of Sciences, Pyongyang, DPR Korea e-mail: mah.ins@sar-co.ne.kp Absrac For 2-D Sokes mixed boundary value problems we consruc a boundary inegral equaion which couples a convenional boundary inegral equaion for he velociy wih a hypersingular boundary inegral equaion for he racion. Expressing erms in he equaion by complex variables, we obain a complex boundary inegral equaion and realize symmerizaion of boundary elemen scheme by Galerkin mehod. Applying a boundary limi mehod, we obain exac calculaion formulae for calculaion of hypersingular boundary inegrals. I is shown ha all divergen erms in hypersingular inegrals cancel each oher ou. Key words: hypersingular inegral, boundary limi mehod, complex boundary elemen mehod, symmeric boundary elemen scheme. AMS 2 subjec classificaion: 65M38, 65P3, 76M5, 78M5, Inroducion Boundary elemen mehodbem allows us o ge approximae soluions of problems of mahemaical physics on a domain Ω in R d d = 2, 3 by he soluion of boundary inegral equaions corresponding o he problems. For large-scale problems, if he final linear sysems are symmeric, hen ieraive mehods are useful, which can reduce he required compuer memory and has a good convergence propery. In view of his for many problems including poenial problems and elasic one, he symmerizaion of boundary elemen scheme is widely considered[9]. For example, o ge a boundary inegral equaion for symmeric boundary elemen scheme o solve poenial problems wih mixed boundary condiions, firs one express he soluion on Ω, poenial, by he sum of simple layer and double layer poenials, densiies of which are boundary flux and boundary poenial. Then, by aking limi o boundary, one ge he poenial boundary inegral equaion. Nex, geing he direcional derivaive of he expression of he poenial and limiing o he boundary again, one ge anoher boundary inegral
S. G. Hong / Elecronic Journal of Boundary Elemens, Vol. 3, No., pp. -32 26 equaion- flux boundary inegral equaion. The flux boundary inegral equaion conains a poenial adjoin o he double layer poenial and a hypersingular boundary poenial[9]. In he case of Dirichle boundary value problem, from he poenial boundary inegral equaion one can ge a simple layer poenial operaor equaion wih he unknown boundary flux. The simple layer poenial operaor is self-adjoin, and so Galerkin discreizaion of he equaion is symmeric. In he case of mixed boundary condiions he velociy boundary inegral equaion includes no only simple layer poenial operaor, bu also a double layer poenial operaor which is no self-adjoin, and so i is difficul o ge symmeric discree scheme from he equaion. Therefore, by combinaion of he poenial boundary inegral equaion wih he flux boundary inegral equaion symmerizaions of convenional poenial problems[, 7] are realized. The flux boundary inegral equaion conains hypersingular inegral, so, i is imporan o ge exac values of hese hypersingular inegrals[9]. For calculaion of hypersingular inegrals Hadamard s finie par inegral is usually used. To apply Hadamard s finie par inegral, basis funcions which become densiy funcions mus be in a leas C,α -class, bu piecewise linear coninuous basis used widely in boundary elemen scheme belong o C, -class. Such siuaions led o sudies of mehods o weaken he singulariy of inegrals[3, 4] and he smoohness requiremen of densiy funcions[6, 2] and o ge numerical values[6, 2, 8]. In [9], in paricular, as a mehod o calculae hypersingular inegral in boundary elemen schemes unlike o Hadamard finie par inegral mehod, he boundary limi mehod was suggesed. Main procedure of he boundary limi mehod is as follows. Firs, wih source poin off he boundary, one inegraes. Then aking limi of he resul as he poin goes o he boundary, one exracs a converging finie par o be regarded as an inegral value and check wheher all divergen erms are canceled. Auhors menioned ha i is an advanage of he boundary limi mehod ha i can obain he value of hypersingular inegral even when densiy funcion is no smooh, as piecewise linear coninuous basis funcion, and deal wih arbirary inegrals wih singulariy by he same way. Bu, for mixed boundary value problems of he Sokes equaion here is no such resuls-symmerizaions and evaluaion of hypersingular inegrals, ye. On he oher hand, he boundary elemen mehod works, commonly, in real variables. For wo dimensional vecor-value problems such as 2-D Sokes flow problem, however, o consider complex variables has advanage ha enable o avoid complexiy due o marix-vecor operaion. Complex boundary elemen mehods were sudied in [3, 4, 2] for poenial problems, in [5] for Helmholz equaion, in [8] for elecrochemical sysem, in [] for flow around obsacles and in [5] for linear elasosaic problems. Using he formulae in [5] for a linear elasic problem, [4] sudied complex poenial expressions for 2-D Sokes flow problems. Bu, he resuls are no for symmeric scheme. 2
S. G. Hong / Elecronic Journal of Boundary Elemens, Vol. 3, No., pp. -32 26 In his paper, we are concerned wih symmeric complex boundary elemen scheme for 2-D Sokes equaion wih velociy and racion boundary condiions ogeher. Following he ideas for poenial and elasic problems above menioned, we have a real boundary inegral equaion o ge symmeric boundary elemen scheme. Expressing variables in he real boundary inegral equaion by complex variables, we obain he complex boundary inegral equaion. We consruc a discree scheme by Galerkin mehod. And assuming exisence of hypersingular complex boundary inegrals on discree boundary, we prove he convergence and sabiliy of he scheme. Furhermore, applying he boundary limi mehod in [9] and proving ha all divergen erms are canceled, we ge formulae for calculaion of all hypersingular complex boundary inegrals. This paper consiss of 5 secions. In Secion 2, combining a velociy boundary inegral equaion and a racion one on he hole boundary, we ge a sysem of velociy equaion on he porion for velociy boundary condiion and racion equaion on he porion for racion boundary condiion, and apply Galerkin scheme o ge symmeric boundary elemen scheme. Then, 5 basic erms are expressed by complex variables and using hem, complex expressions of boundary poenials are obainedtheorem 2.. Afer ha, a complex boundary elemen scheme is consruced by using piecewise linear coninue elemens for he velociy and piecewise consan elemens for he racion. Symmery, convergence and sabiliy of he scheme are provedtheorem 2.2. In Secion 3, by he boundary limi mehod we ge he exac formulae for calculaion of hypersingular complex boundary inegrals in he scheme and prove cancelaion of all divergen ermstheorem 3.-3.5. In Secion 4 an algorihm o calculae all elemens excluding hypersingular boundary inegral in coefficien marix of he complex discree equaion and elemens in righ-hand side of he equaion. In Secion 5, resuls of he paper are summarized. Finally, many proofs and calculaions which make he reading of he paper difficul are colleced in Appendix. 2 Symmeric complex boundary elemen scheme 2. Sokes mixed boundary value problem Le us consider he following mixed boundary value problem of he Sokes equaion. µ u + p =, divu =, x Ω R 2, u = f, x, 2 T u = g, x 2. 3 Here Ω is a simply conneced bounded domain and boundary = 2 is smooh, u he velociy, p he pressure, µ he dynamical viscociy coefficien,f 3
S. G. Hong / Elecronic Journal of Boundary Elemens, Vol. 3, No., pp. -32 26 andg are, respecively, races of velociyu and raciont u on and 2, and n is ouward ouward normal uni vecor on he boundary of domain. Tracion is given by T u := pn + µ u + u T n 4 and i s adjoin racion is defined by T u := pn + µ u + u T n. 5 Le Ex, y be he fundamenal soluionmarix o he Sokes operaor. Define F x, y := T yex, y T, Gx, y := T x Ex, y, Hx, y := T x T yex, y T. Denoe by V, K, K and D, respecively, boundary inegral operaors of which kernels are, respecively, Ex, y, F x, y, Gx, y and Hx, y. Then, by he boundary inegral operaors we have he following coninuous mappings[]. V : H 2 H 2, K : H 2 H 2, K : H 2 H 2, D : H 2 H 2. Wih ensor noaion, kernel funcions can be wrien as follows[, 5]. E i,j x, y = δ ij log 4πµ r + r,ir,j 2 δ ij, 6 F i,j x, y = πr r,ir,j r,k n k y, 7 G i,j x, y = πr r,ir,j r,k n k x, 8 H i,j x, y = µ µ x, y = πr2 πr 2 [δ ijr,k r,k r,l n l y 9 + n i yr,j r,k + n k yr,i r,j + δ ik n j ]n k x, where r = x y, r,j = r/ y j and Einsein convenion for sub-indices is used. Consan erm 2 δ ij in expression 6, which does no affec he soluion, is added for convenience in he complex variable expression. Puing u =: φ, T u =: τ, we obain boundary inegral equaion[] φx = V τx Kφx, 2 x, 2 τx = K τx + Dφx, x. 4
S. G. Hong / Elecronic Journal of Boundary Elemens, Vol. 3, No., pp. -32 26 Le us inroduce funcional spaces as follows. s : H s i := {v i : v H s }, H s i := complee {v H s : suppv i }, s < : H s i := H s i, Hs i := H s i. Denoe H s j H s i conracions of he operaors V, K, K and D, respecively, by V i,j, K i,j, K i,j and D i,j. Seing T u = w, u 2 = ϕ and aking ino accoun he boundary condiion of he firs boundary inegral equaion ax and one of he second equaion ax 2, we ge he following boundary inegral equaion 2 fx + K,fx + K,2 ϕx V, wx V,2 gx =, x, 2 gx K 2,wx K 2,2gx D 2, fx D 2,2 ϕx =, x 2, which is wrien as follow. [ ] [ ] [ ] [ ] V, K,2 w V =,2 /2I + K, g D 2,2 ϕ /2I 2 K 2,2 D 2, f K 2, where I i, i =, 2 are,respecively, ideniy in i. Le V := H 2 H 2, Ṽ := H 2 H 2 2 [ ] τ and rewrie V φ as [ ] τ = φ [ τ φ ] + ] [ τ, φ [ ] τ Ṽ, φ ] [ τ V φ. Define a bilinear form [ ] [ ] [ ] [ ] [ ] τ ξ V K τ ξ a φ, := p K D φ,, Ṽ p Ṽ R. Then, by we have [ ] [ ] [ ] τ ξ ξ a φ, = b p p where b, [ ] [ ξ V := p /2I K D [ ] ξ p Ṽ, /2I + K ] ] [ ] [ τ ξ,. 2 φ p 5
S. G. Hong / Elecronic Journal of Boundary Elemens, Vol. 3, No., pp. -32 26 Now, we conver real vecor poenials V τx, Kφx, K τx and Dφx o complex variable forms. To his end, pu ς := x + ix 2, z := y + iy 2, nz = n y + in 2 y, φz := φ y + iφ 2 y, τz := τ y + iτ 2 y. Then, we have ha x y = 2 ς z + ς z, x 2 y 2 = ς z ς z 2i and componens of vecor value funcion u y, u 2 y T are as follows. u y = uz + uz, u 2 y = uz uz. 2 2i Also we have he following equaliies which are proved in A of he Appendix. i r,j x, yu j y = 2r ς zuz + ς zuz, ii n j yu j y = 2 nzuz + nzuz, iii r,j x, yn j x = 2r ς znς + ς znς, iv r,j x, yn j y = 2r ς znz + ς znz, v r,i x, yr,j x, yu j y = ς z 2 ς z. uz + uz Theorem 2. The followings equaliies hold. Ex, yτydy = E,j x, yτ j ydy + i E 2,j x, yτ j ydy = 8πµ F x, yφydy = = 4π log ς z τz + log ς z τz + ς z ς z τz F,j x, yφ j ydy + i ς z nzφz + ς z ς z 2 nzφz dz, F 2,j x, yφ j ydy + nzφz + nzφz dz, ς z Gx, yτydy = G,j x, yτ j ydy + i G 2,j x, yτ j ydy = 4π + ς z τz + ς z τz ς z τz + ς z ς z 2 τz nς nς dz, 3 4 5 6
S. G. Hong / Elecronic Journal of Boundary Elemens, Vol. 3, No., pp. -32 26 Hx, yφydy = = µ 2π + H,j x, yφ j ydy + i ς z 2 nzφz + ς z 2 nzφz ς z 2 nzφz + nzφz + H 2,j x, yφ j ydy nς 2ς z ς z 3 nzφz nς dz. 6 Proof. Now using formulae of kernel funcions 6-9 and equaliiesi-v, we have ha E i,j x, yτ j y = = τz 4πµ 2 log ς z ς z + 2 log = 8πµ ς z ς z ς z τz + log ς z τz + ς z ς z τz τz + τz, τz 2 F i,j, yφ j y = = ς z φz + φz ς znz + ς znz πr 2 ς z 2 ς z = ς z φz + φz 4π ς z ς z 2 ς znz + ς znz = ς z nz φz + φz 4π ς z ς z + nz ς z = 4π nzφz ς z G i,j x, yτ j y = = ς z πr 2 ς z = nςτz 4π ς z + nzφz + nzφz ς z τz + τz 2r Le us rewrie he 9 as follows ς znς + ς znς + nςτz + nςτz ς z + ς z ς z 2 nzφz, + ς z ς z 2 nςτz. H i,j x, yφ j y = µ πr 2 [φ ir,k n k r,l n l + r,i φ k n k r,l n l 8r,i r,j φ j r,k n k r,l n l + n i r,j φ j r,k n k + r,i n k n k r,j φ j + n i n j φ j ]. By i-v erms in [ ] of he righ hand side are expressed as follows. 7
S. G. Hong / Elecronic Journal of Boundary Elemens, Vol. 3, No., pp. -32 26 φ i r,k n k r,l n l = = φz 2r ς znς + ς znς ς z nz + ς z nz 2r = nς 4r 2 ς z ς z φ z nz + ς z ς z φ z nz + nς 4r 2 ς z ς z φ z nz + ς z ς z φ z nz, r,i φ k n k r,l n l = ς z 2r φznς + φznς ς znz + ς znz = 2r = nς 4r 2 ς z ς zφznz + ς zς zφznz + nς 4r 2 ς z ς zφznz + ς zς zφznz 8r,i r,j φ j r,k n k r,l n l = 8 ς z ς zφz + ς zφz r 2r ς znς + ς znς ς znz + ς znz 2r 2r = 4r 2 nς 4 r 2 ς z 2 φz + ς z ς zφz ς z 2 nz + ς zς znz + 4r 2 nς 4 r 2 ς z 2 φz + ς z ς zφz ς z ς znz + ς z 2 nz = ς z 4r 2 nς4 φz + φz ς z 2 nz + ς zς znz ς z + ς z 4r 2 nς4 φz + φz ς z ς znz + ς z nz 2, ς z, n i r,j φ j r,k n k = = nz 2r ς zφz + ς zφz ς znς + ς znς 2r = nς 4r 2 ς z ς zφznz + ς z ς zφznz + nς 4r 2 ς zς zφznz + ς z ς zφznz, 8
S. G. Hong / Elecronic Journal of Boundary Elemens, Vol. 3, No., pp. -32 26 r,i n k n k r,j φ j = = z ς 2r nznς + nznς ς zφz + ς zφz 2r = nς 4r 2 ς zς zφznz + ς z ς zφznz + nς 4r 2 ς zς zφznz + ς z ς zφznz n i n j φ j = nς nzφz + nzφz 2 = 4r 2 nςς z ς z2 nzφz + nzφz. Therefore, H i,j x, yφ j y = = µ nς { ς z ς zφznz + ς z ς zφznz 4πr4 + ς z ς zφznz + ς zς zφznz 4ς z ς zφznz 4ς z 2 φznz 4 ς z 2 φznz 4 ς zς zφznz + ς z ς zφznz + ς z ς zφznz + ς zς zφznz + ς z ς zφznz + 2ς z ς zφznz + 2ς z ς zφznz } + µ nς { ς z ς z φ z nz + ς z ς z φ z nz 4πr4 + ς z ς zφznz + ς zς zφznz 4 ς z 2 ς z3 φznz 4 ς z φznz 4 ς z ς zφznz 4 ς z 2 φznz + ς z ς zφzn z + ς z ς zφzn z + ς zς zφznz + ς z ς zφznz } 9
S. G. Hong / Elecronic Journal of Boundary Elemens, Vol. 3, No., pp. -32 26 { = µ 2πr 4 ς z 2 φznz + ς z 2 φznz nς } + ς z 2 ς z3 φznz + φznz + 2 ς z φznz nς { = µ 2π ς z 2 φznz + ς z 2 φznz nς } 2ς z + ς z 2 φznz + φznz + ς z 3 φznz nς. This finishes he proof of he heorem. 2.2 Symmeric complex boundary elemen scheme We suppose ha h = h 2h is quasi-uniform discreizaion of and N h = [z s, z s+ ], 2h = s= N 2 = [, + ], N + N 2 =: N, 7 where he order of nodes is pu along he curve as couner-clockwise. boundary elemen spaces Define S h := S h S 2 h Ṽ, 8 S h := span {w s : suppw s h, s =,, N }, S 2 h := span {ϕ : suppϕ 2h, =,, N 2 }. Here basis funcions are { z [zs, z w s z = s+ ] z / [z s, z s+ ], 9 z/ z [, ] ϕ z = + z/+ z [, + ] z / [, ] [, + ] Then, we ge a discree equaion based on Galerkin mehod [ ] [ ] [ ] Vh K h τ h ξh K h D h φ, = h p h [ ] ] [ ] [ ] V = h /2I + K h [ˆτ ξh ξh /2I K h, S D h ˆφ h, p h. 2 p h 2
S. G. Hong / Elecronic Journal of Boundary Elemens, Vol. 3, No., pp. -32 26 where V h, K h, K h, D h, which are discreizaions of V, K, K, D are defined on [ ] τ space S h, and he unknown vecor is h Sh := Ṽ S h. Puing τh = N α s w s and φ h = N 2 N s= N = φ h β ϕ, we ge a sysem of equaions equivalen o 2 V h α s w s, w s N 2 = K h β ϕ, w s = ˆb, ws s =,, N, K h α sw s, ϕ + N 2 D h β ϕ, ϕ = ĉ, ϕ =,, N 2. s= = The poenials in he lef hand side of 22 are calculaed, respecively, by he poenial expressions of 3-6 on h. In real variable, 22 are he sysem of 2Norder. We spli he 2N 2N coefficien marix ino four blocks by V h, K h, K h, D h. Then we ge Theorem 2.2 Assume ha a hypersigular inegral in he lef [ hand ] side of τ has a finie value. Then equaion has a unique soluion Ṽ. Assume ha oal of hypersigular inegrals of every elemen in D h -blocks has finie value. Then, V h and D h -blocks are, respecively, symmeric, K h and [ K h] -blocks τ are block skewsymmeric each oher, and 2 has a unique soluion h φ Sh. [ ] h τ Furhermore, if condiion φ H /2+σ H /2+σ 2, σ < 2 holds, hen we have error esimaion [ ] [ τ τ φ h φh] φ [ ] CN s τ H φ, s H s s= 22 H H 2 s σ + 23 2. Proof. The operaor V is a self-adjoin operaor saisfying V τ, τ L2 cv τ 2 H 2, τ H 2, c V >, 24 and he operaor D is a self-adjoin operaor saisfying Dφ, φ L2 cd 2 φ 2 H 2, φ H 2 / kerd, c D 2 >. 25 In he case of linear elasic poenial operaors his resul is already proved Theorem 2 in [7]. Regarding he Sokes flow poenial operaor as special case of linear elasic poenial operaors wih ν = 2, we ge 24, 25. Since Kφ, τ + φ, K τ = φ, K τ + φ, K τ =,
S. G. Hong / Elecronic Journal of Boundary Elemens, Vol. 3, No., pp. -32 26 we have [ ] [ ] [ ] τ τ a, c τ 2 φ φ φ Ṽ [ ] τ φ Ṽ, where c = min { } c V, c D 2. By Lax-Milgram heorem we ge unique exisence of a soluion o and in he same way we ge he soluion for 2. We can ge he error esimaion of he 23 exending he resuls of Theorem in [7] for scalar o vecor space. The symmery of marix-blocks will be described in deail in A2 of he he Appendix. 3 Analyical calculaion of hypersingular inegrals In 22, inegrals V h α s w s, w s and K h β ϕ, w s, K h α sw s, ϕ conain weak singular inegrals, and D h β ϕ, ϕ have hypersingular inegrals. 3. Calculaion of hypersingular boundary inegral by boundary limi mehod In his subsecion we apply he boundary limi mehod in [9] o hypersingular boundary inegrals given on he boundary of domain Ω in complex plane C. In he boundary elemen scheme in R d d = 2, 3, le us consider he definiion of hypersingular boundary inegralcf..2.6 in [] Dux = lim x x,x x Ω n x E n y x, yuyd y. To be clear, we assume a poin x be on he normal line passing poin x. Since x Ω, y, i implies x y >, herefore x n x E x, yuydy = n y 2 E n x n y x, yuydy because E n y x, y is smooh and so differenial symbol x n x = enered ino inegral symbol. Puing Hx, y := 2 E n x n y x, y, we ge Dux = lim x x,x Ω n x can be Hx, yuydy. 26 Denoing x x = ε, we have x = x n x ε, and Dux = lim Hx n x ε, yuyd y. 27 ε 2
S. G. Hong / Elecronic Journal of Boundary Elemens, Vol. 3, No., pp. -32 26 Formula 27 is boundary limi scheme in [9]. In he case of complex variable, definiion 26 of hypersingular boundary inegral can be wrien as follows Duς = lim Ω ς ς Hς, zuzdz. 28 For simpliciy of noaion, we used he same funcion symbols wihou change. Assume a poin ς is on he normal line passing he poin ς and nς = nς. If he poin ς be away ε ouside from he boundary, hen ς = ς + εn ς and Duς = lim H ς + εnς, z uzdz. 29 ε Denoing by ς he angen vecor wih couner-clockwise along he boundary curve a he poin ς, we have nς = iς, and so 26 may wrien as follows. Duς = lim ε H ς iες, z uzdz. 3 As we see, boundary limi mehod requires only inegrabiliy of densiy uy. Therefore, when uz is coninue we can apply boundary limi mehod. Sign - before iε in 3 means approach process from ouside o boundary. In conras, sign + means from inside o boundary. 3.2 Calculaion of hypersingular marix elemen As menioned a he beginning of his secion, hypersingular inegrals appear in calculaion of elemens of block marices D h β ϕ, ϕ,, =,..., N 2. In more deail, i is only when suppors of basis funcions ϕ and ϕ are overlapped, ha is, = suppors are coincided compleely or = suppors are overlapped by half Le and e := +, L := e, =,..., N 2 3 z = + ξe, z [, + ], ξ, ς = + ηe, ς [, + ], η. 32 Then, we ge { ξ z [zs, z ϕ z = ] ξ, z [, + ], 33 nz = ie /L, dz = L dξ, z [, + ]. 34 3
S. G. Hong / Elecronic Journal of Boundary Elemens, Vol. 3, No., pp. -32 26 3.2. Calculaion of marix elemen D h β ϕ, ϕ, = Since supp ϕ z = supp ϕ ς = [, ] [, + ], inerior inegral variable z and exerior inegral variable ς of muliple inegral z + z + dς dz are me each oher in inegral process, and he inegral is hypersingular. Using expression 6 of hypersingular poenial, we have D h β ϕ, ϕ = µ 2π + ϕ ςdς + + ς z 2 nzβ ϕ z + nzβ ϕ z = µ 2π β µ 2π β + + ϕ ςdς ϕ ςdς + + nzβ ϕ z ς z 2 + nzβ ϕ z ς z 2 nς + 2ς z ς z 3 nzβ ϕ z nς dz ς z 2 nznς + ς z 2 nznς ϕ zdz nznς + nznς 2ς z ς z 2 + ς z 3 nznς ϕ zdz. Spliing he las wo inegral erms ino elemens suppressing µ 2π and inegrands, we have where β z + z + z + β = β +β + +β =: I β + I 2 β + I 3 β + I 4 β, z + = β + β + + β =: I β + I 2 β + I 3 β + I 4 β. Finally, we obain z + z + +β + β z + z + z + z + 35 36 37 D h β ϕ, ϕ = µ 2π I µ 2π I 2 µ 2π I 3 µ 2π I 4, 38 I k = I k β + I k β, k =, 2, 3, 4. 39 4
S. G. Hong / Elecronic Journal of Boundary Elemens, Vol. 3, No., pp. -32 26 All hese inegrals are calculaed in A4. of Appendix. Among above inegrals only I β, I 2 β 2, I 3 β 3, I 4 β 4 have, respecively, hypersingulariy which imply appearance of divergen erms. Bu following wo heorems shown his divergen erms cancel each oher ou. Firs, from he compuaion of I β and I 2 β we ge following Theorem 3. The firs wo inegrals µ 2π I and µ 2π I 2 in calculaion of elemen D h β ϕ, ϕ have, respecively, divergen erms µ 2π β logε 2 and µ 2π β logε 2. They are cancelled in he sum µ 2π I µ 2π I 2 and we ge finie inegral value µ 2π I µ 2π I 2 = β µ e loge 2 + e loge 4π e e + 2 + e + e loge + e ē logē 2 + ē logē e e ē ē + 2 + ē + ē logē + ē ē ē + β µ 3 e + e e ē 4π ē ē ē 2 e logē ē e + e ē e ē ē ē 2 logē ē 2 e e + e ē ē ē ē 2 logē + ē. Also from he compuaion of I 3 β and I 4 β we ge following Theorem 3.2 Las wo inegrals µ 2π I 3 and µ 2π I 4 of 38 in calculaion of hypersingular elemen D h β ϕ, ϕ have, respecively, divergen erms µ 2π β logε 2 and µ 2π β logε 2. These are cancelled in sum µ 2π I 3 µ 2π I 4 and we ge he following finie inegral value. µ 2π I 3 µ 2π I 4 = β µ 4π e loge 2 + e e e ē logē 2 + ē logē + ē ē + β µ 3 e + e e ē 4π ē ē ē 2 e ē logē + ē 2 loge + 2 + ē e ē e e + e ē ē ē ē 2 4 2 + e + e loge + e e e + ē logē + ē ē ē e logē + e ē ē logē + ē. ē 2 4 5
S. G. Hong / Elecronic Journal of Boundary Elemens, Vol. 3, No., pp. -32 26 From all inegrals of 39, which are calculaed in A4., we have Theorem 3.3 Divergen erms being canceled, hypersingular elemens D h β ϕ, ϕ, =,..., N2, have he following finie inegral value. D h β ϕ, ϕ = = β µ + e 2π e + 2 + e + e e + ē + β µ + e ē e ē ē 2 loge + e + ē loge + e e logē + 2 + ē + ē ē ē e 2 + e e + 2π ē ē ē e ē logē + ē 2 ē e ē ē 2 e loge logē ē logē + ē logē e ē e ē + e ē ē 2 3.2.2 Calculaion of marix elemen D h β ϕ, ϕ, = Wihou loss of generaliy, suppose ha = +. Then suppϕ z suppϕ ς = [, + ]. By 6 we ge logē + ē. D h β ϕ, ϕ = D h β ϕ, ϕ + = = µ z +2 z + 2π β nznς ϕ + ςdς ς z 2 + nznς ς z 2 ϕ zdz µ z +2 2π β ϕ + ςdς ϕ zdz. + 42 nznς + nznς 2ς z ς z 2 + ς z 3 nznς As 36 and 37 le us express wo erms in he righ hand side above formula 6
S. G. Hong / Elecronic Journal of Boundary Elemens, Vol. 3, No., pp. -32 26 as follows. β z +2 z + = β z + +β z + + +β z +2 +β z +2 z + z +2 β z + = β z + + + =: I β + I 2 β + I 3 β + I 4 β, + β z + + + β z +2 + β z +2 z + 43 + + =: I β + I 2 β + I 3 β + I 4 β. Among above inegrals only I 2 β 2 has divergen erms,bu his divergen erms cancel each oher ouappendix:a4.2., so we ge Theorem 3.4 For every pair, such ha =, elemen D h β ϕ, ϕ, canceling divergen erms each oher, has a finie inegral value. As shown above, elemens of boundary elemen marix be calculaed by various ypes of inegrals, where he elemen corresponding o hypersingular inegral can have divergen erms. If here exis a elemen ha i s divergen erms are no cancelled, he scheme is useless. From his poin of view we define he compaibiliy of he scheme. Definiion 3. When for every elemen deermined by hypersingular inegral among elemens of discree marices obained in boundary elemen scheme, all divergen erms are cancelled each oher, we say he scheme have compaibiliy. Theorem 3.5 Galerkin scheme 22 has compaibiliy. Proof. In he marix calculaion of he lef hand side in 22 divergen erms are from D h -block. In deail, divergen erms are from he hypersingular inegral elemen D h β ϕ, ϕ, = in which suppors of basis funcions in inerior and exerior inegral are coincided compleely and he hypersingular inegral elemen D h β ϕ, ϕ, = in which he suppors are coincided by half, and no from ohers in D h -block. Theorem 3.3 shows ha divergen erms from hypersingular inegral D h β ϕ, ϕ, =, are cancelled and Theorem 3.4 does he same for he case of hypersingular inegral D h β ϕ, ϕ, =. Since all divergen erms are cancelled, he scheme has compaibiliy. 4 Algorihm Sep. Consrucion of complex discree equaion In he marix calculaion of he final discree equaion 22 calculaion of V h - block elemens can be fulfilled by simple analyical calculaion because operaor V h is weak singular and basis funcions w s, w s are consans. 7
S. G. Hong / Elecronic Journal of Boundary Elemens, Vol. 3, No., pp. -32 26 For elemens of K h, K h -blocks, suppors of corresponding basis funcions ϕ, w s and w s, ϕ have no subsecion, which allow us o calculae hem by quadraure. For elemens of D h - blocks, since in he case of > suppors of basis funcions ϕ, ϕ have no subsecion or only one poin as subsecion, i can be calculaed by quadraure or simple analyical inegral. Righ hand side of discree equaion 22 may be wrien as follows. b, ws := V h τ, w s + 2 φ, w s + K h φ, w s, ĉ, ϕ := 2 τ, ϕ K h τ, ϕ D 44 h φ, ϕ, ] [ τ where V φ is arbirarily aken wih condiion τ 2 = g, φ = f. [ ] τh We ake an approximaion V φ h as follows. h N N 2 N N 2 τ h = τ s w s + τ N +ϕ, φ h = φ s w s + φ N +ϕ, 45 s= = s= where τ s, φ s, s N elemen mean, τ N +, φ N + N 2 node value. Then, we ge = V h τ, w s = N s= V h τ s w s, w s + N2 = V h τ N+ϕ, w s, K h φ, w s = N K h φ s w s, w s + N2 K h φ N+ϕ, w s, s= = K h τ, ϕ N = K h τ sw s, ϕ + N2 K h τ N +ϕ, ϕ, s= = D h φ, ϕ = N D h φ s w s, ϕ + N 2 D h φ N +ϕ, ϕ, s= = 46 φ, w s = φ s, τ, ϕ = +2 = 2 τ N+ ϕ, ϕ. 47 Since he coefficiens τ s, τ N+, φ s, φ N+ are known, he inegrals above can be calculaed as before. Sep 2. calculaion of real discree equaion Calculaing every block of lef hand side in 22, we ge a complex discree equaion for s =,..., N and =,..., N 2 [ ] [ ] [ ] [ ] As B αs Es F + αs = C s D β G s H β [ ] Rs, S s, Denoing complex marices and vecors by A s = A s +A 2 s i and α s = α s +iα 2 s, 48 8
S. G. Hong / Elecronic Journal of Boundary Elemens, Vol. 3, No., pp. -32 26 we have he following real discree equaion A s + E s A 2 s + E s 2 B + F C s + G s C s 2 + G 2 s D + H A 2 s + E s 2 A s E s B 2 + F 2 C s 2 + G 2 s C s G s D 2 + H 2 [ ] T = R s, S s, R 2 s, S 2 s,. B 2 D 2 B D + F 2 + H 2 F H α s α s 2 β β 2 = 49 Denoing again he oal unknown vecor by T α, α2,..., α N, α 2 N, β, β2,..., β N 2, β 2 N 2 = α, α 2,..., α 2N, α 2N +,..., α 2N T =: X assembling o his vecor we combine he oal coefficien marix as following a,... a,2n a,2n +... a,2n.................. A = a 2N,... a 2N,2N a 2N,2N +... a 2N,2N a 2N+,... a 2N+,2N a 2N+,2N +... a 2N+,2N.................. 5 a 2N,... a 2N,2N a 2N,2N+... a 2N,2N [ ] AVh A = Kh A K h A Dh and denoe he righ hand side by B. Then We ge he resuling equaion AX = B. 5 Noes ha In marix A block A Vh and block A Dh are symmeric,respecively, A Kh and A K h is skewsymmeric. By solving his equaion, we can direcly have he racion and he velociy on he boundary, ha is, α s + τ s, α s 2 + τ s 2 racion for elemen [z s, z s+ ] h, β + φ N, + β2 + φ 2 N + velociy a node 2h. Sep 3. Calculaion of velociy and pressure in domain To ge he velociy and pressure a he required posiion in domain, we can use he formula uς = V h τ + ας K h φ + βς, pς = Q h τ + ας + R h φ + βς, ς Ω, 52 9
S. G. Hong / Elecronic Journal of Boundary Elemens, Vol. 3, No., pp. -32 26 where Q h τς = 4π R h φς= µ 2π Λ ς z τz + ς z τz dz, ς z 2 nzφz + ς z 2 nzφz dz, 53 α s = 5 Conclusion { αs, s N,, N < s N, β = {, N, β N, N < N. 54 In he paper, we realize symmeric complex boundary elemen scheme of 2-D Sokes mixed boundary value problem and proved he convergence, sabiliy of he scheme. In paricular, we presened he mehod o conver boundary elemen scheme ino complex boundary elemen scheme direcly. We go he analyical calculaion formulae for hypersingular boundary inegral in he scheme by boundary limi mehod, o do his, we proved cancelaion properies of he divergen erms in hypersingular inegrals. Appendix A. Proof of i-v in Subsecion 2. i : r,j x, yu j y = r x y u y + x 2 y 2 u 2 y = r 2 ς z + ς zu y + 2i ς z ς zu 2y = 2r ς zu y iu 2 y + ς zu y + iu 2 y = ς zuz + ς zuz. 2r ii : n j yu j y = n yu y + n 2 yu 2 y = 2 nz + nzu y + 2i nz nzu 2y = nzu y iu 2 y + nzu y + iu 2 y 2 = nzuz + nzuz. 2 iii, iv are proved similarly o i. 2
S. G. Hong / Elecronic Journal of Boundary Elemens, Vol. 3, No., pp. -32 26 v : r,i x, yr,j x, yu j y = = ς z ς zuz + ς zuz r 2r = ς z 2r 2 ς zuz + ς zuz = ς zuz + ς zuz 2 ς z = ς z uz + uz. 2 ς z A2. Proof of Symmery of he Real Coefficien Marix of he Sysem in 22 Denoe suppors of basis in h by π,..., π s,..., π N and in 2h by π,..., π,..., π N2, respecively. Noe ha π s is elemen, π is sum of wo elemens which has common poin. Subscrips i, j of enries of he coefficien marix are define as following. { 2s i =, π = π s h, s =,..., N 2N + 2 π = π 2h,, =,..., N 2 j = { 2s, π = πs h, s =,..., N 2N + 2 π = π 2h, =,..., N 2. Noe ha i, j =,..., 2N, 2N +,..., 2N. Firs, we will esablish symmery of V h.-block V h α s w s, w s = w s ςdς 8πµ log = 8πµ π s ς z α sw s z + log dς π s log ς z α sw s z + ς z ς z α sw s z dz ς z α s + log ς z α s + ς z ς z α s dz. 2
S. G. Hong / Elecronic Journal of Boundary Elemens, Vol. 3, No., pp. -32 26 By Theorem 2., in real variable, V h α s w s, w s = dx log 4πµ x y I 2 π s π s [ ] ] x + y 2 x y x 2 y 2 [α s x y 2 x 2 y 2 x y x 2 y 2 2 dy α s 2 [ ] [ ] a;s s a =: 2;s s α s a 2;s s a 22;s s α s 2 ; a ;s s = 4πµ π s a 2;s s = 4πµ π s a 2;s s = 4πµ π s a 22;s s = 4πµ π s dx π s dx π s dx π s dx π s log x y 2 + x y 2 x y 2 dy x y x 2 y 2 x y 2 dy, x 2 y 2 x y x y 2 dy = a 2;s s, log x y 2 + x 2 y 2 2 x y 2 dy. [ ] a;s s a Obviously, elemen-marix 2;s s is symmeric. On he oher hand, we ge a ;ss = dx 4πµ π s π s = dy 4πµ π s π s = dx 4πµ π s π s = dy 4πµ π s π s a 2;s s a 22;s s log x y 2 + x y 2 x y 2 dy log x y 2 + x y 2 x y 2 dx log y x 2 + y x 2 y x 2 dy log x y 2 + x y 2 x y 2 dx = a ;s s Similarly, a 2;ss = a 2;s s, a 2;ss = a 2;s s, a 22;ss = a 22;s s, herefore [ ] [ ] a;s s a 2;s s a;ss a = 2;ss. a 2;ss a 22;ss a 2;s s a 22;s s 22
S. G. Hong / Elecronic Journal of Boundary Elemens, Vol. 3, No., pp. -32 26 For i := 2s, j := 2s, s, s =,..., N, expressing as [ ] [ ] a;s s a 2;s s ai,j a =: i,j+ a i+,j a i+,j+ a 2;s s a 22;s s we have [ ] [ ] [ ] T ai,j a i,j+ ai,j a = i+,j ai,j a = i,j+ a i+,j a i+,j+ a i,j+ a i+,j+ a i+,j a i+,j+ Le us express V h -block by A Vh. Then A Vh = a i,j = a j,i = A T V h Similarly, symmery of D h -block is esablished. i.e. A Dh = A T D h. Now we will esablish block skewsymmery of blocks K h and K h, i.e. A K h = A T K h. K h β ϕ, w s = w s ςdς 4π + ς z = 4π ς z nzβ ϕ z + π s ς z ς z 2 nzβ ϕ z nzβ ϕ z + nzβ ϕ z dz dς π ς z nzβ ϕ z + ς z ς z 2 nzβ ϕ z + nzβ ϕ z + nzβ ϕ z dz ς z = dx πr r,ir,j β j r,k n k yϕ ydy π s = π s dx π π [ x y x y x 2 y 2 x y x y x 2 y 2 x 2 y 2 x 2 y 2 x y nyϕ y π x y 4 dy =: [ ] b;s b 2;s. b 2;s b 22;s ] [ β [ ] b;s b Therefore. elemen-marix of K h -block in 22 is 2;s and, obviously, symmeric. b 2;s β 2 ] b 22;s 23
S. G. Hong / Elecronic Journal of Boundary Elemens, Vol. 3, No., pp. -32 26 Similarly, we ge for K h -block K hα s w s, ϕ = π x y nx π x y 4 dy =: ϕ xdx π s [ [ a; s a 2; s a 2; s a 22; s x y 2 ] x y x 2 y 2 x 2 y 2 x y x 2 y 2 2 ] [ ] α s α s 2. Here a ; s = ϕ xdx x y 2 x y nx π x y 4 dy π π s 2 x y nx = dy x y π x y 4 ϕ xdx similarly, π s π s π 2 y x ny = dx y x π y x 4 ϕ ydy = dx π s π π 2 x y ny x y π x y 4 ϕ ydy = b ;s, a 2; s = b 2;s, a 2; s = b 2;s, a 22; s = b 22;s Therefore elemen-marix [ ] [ ] a; s a 2; s b;s b = 2;s b 2;s b 22;s a 2; s a 22; s is symmeric. We denoe for i = 2s, j = 2N + 2, s =,..., N, =,..., N 2 [ ] [ ] b;s b 2;s ai,j a =: i,j+ b 2;s b 22;s a i+,j a i+,j+ a and for i = 2N + 2, j = 2s, s =,..., N, =,..., N 2 [ ] [ a; s a 2; s ai,j =: a ] i,j + a 2; s a 22; s a i +,j a. a2 i +,j + Then elemen-marices a and a2 are symmeric, respecively, moreover if s = s, = hen i = 2s = j, j = 2N + 2 = i and hus [ ] [ ] [ ] b;s b 2;s b;s b = 2;s ai,j a = i,j+, b 2;s b 22;s a i+,j a i+,j+ b 2;s b 22;s 24
S. G. Hong / Elecronic Journal of Boundary Elemens, Vol. 3, No., pp. -32 26 [ ] [ a; s a 2; s ai =,j a ] [ i,j + aj,i a a 2; s a 22; s a i +,j a = j,i+ i +,j + a j+,i [ ai,j a For all i, j combining elemen-marices i,j+ and combining elemen-marices a i+,j a i+,j+ a j+,i+ ]. ] we ge block A Kh, [ ] aj,i a j,i+ block A a j+,i a T K From a2 we j+,i+ h. have A K = A T K. [ ] AVh A Collecing resuls we have ha in he real coefficien marix A := Kh A K h A Dh blocks A Vh and A Dh are, respecively, symmeric, and A Kh and A K h are block skewsymmeric. A3. Explain on Mapping properies of he boundary inegral Operaors For Sokes flow problem i was proved ha mappings of he boundary inegral operaors V : H /2 H /2, K : H /2 H /2, D : H /2 H /2 are coninuouscf. [], Lemma 5.6.4, where he operaors V, K, D are denoed γ V, γ W, D, respecively. For elasosaic problem which correspond o he Sokes problem when he Poisson raio ends o /2, i holds ha K : H /2 H /2 is coninuouscf. [6], Theorem 2 for s =. These properies are obained by using solvabiliy and regulariy of associaed ransmission problemscf. [], 5.6.2. Noe ha for he poenial problem also above properies are held. In he case of 2-D poenial problem we explain only for simple layer boundary inegral operaor V, he inegral kernel of which is π log x y refer o [2] Chaper 7. Denoing by C he circumsance of uni circle abou he origin, le us inroduce 2πperiodic parameer as x = cos, sin, y = cos s, sin s, x, y C,, s [, 2π]. Now, choose arbirary densiy σy L 2 C. Then σy = σcos s, sin s =: φs and V σx = V φ = 2π = π = π 2π 2π log log =: V φs + V φs, 2π log sin 2 s 2 /2 φsds = π φsds cos cos s 2 + sin sin s 2 /2 2π 2e /2 sin s 2 φsds + π log 2e /2 2e /2 sin s 2 φsds 2π log2 /2φsds 25
S. G. Hong / Elecronic Journal of Boundary Elemens, Vol. 3, No., pp. -32 26 where V is principal operaor and, obviously, V φ and V φ are,respecively, 2π periodic funcions and belong o L 2, 2π. For any real number r, define H r 2π o be he se of all 2π periodic funcions φ L 2, 2π for which φ r = [ ˆφ 2 + 2π m > m 2r ˆφm 2 ] /2 <, where ˆφm is Fourier coefficien of φs, ha is ˆφm = 2π φse ims ds. 2π I holds cf. [2],7.2.74 ha V φ = ˆφ + 2π m > ˆφm e. im m Then, for r i holds ha V φ r+ = [ ˆφ 2 + 2π = [ ˆφ 2 + 2π which implies ha m > m > m 2r+2 ˆφm ] /2 2 m m 2r ˆφm 2 ] /2 = φ r, V : H r 2π H r+ 2π isomorphism. This esimae is exended o allow r < by dualiy, herefor, for r = /2 give V : H /2 2π H /2 2π, and, aking accoun of V φ = cons, V : H /2 2π H /2 2π. This resul lead o V : H /2 C H /2 C, moreover, for smooh boundary V : H /2 H /2, 26
S. G. Hong / Elecronic Journal of Boundary Elemens, Vol. 3, No., pp. -32 26 which all are coninuous. A4. Calculaion of hypersingular inegrals A4.. Calculaion of marix elemen D h β ϕ, ϕ, = Firs we calculae he inegrals I β, I 2 β. I β = = β ϕ ςdς ς z 2 nznς + ς z 2 nznς ϕ zdz. a3 By 32-34 in subsecion 3.2 we have z = + ξe, ϕ z = ξ, nz = ie /L, dz = L dξ, ς = + ηe, ϕ ς = η, nς = ie /L, dς = L dη. a4 Now, le us apply boundary limi mehod for calculaion of I β. Taking a poin ς ε = ς + εl nς apar ς, by variable change a4 we have ς ε = + ηe + εl ie /L = + η iεe, ς ε z = + η iεe + ξe = ξe + η iεe. Denoe by I ε β he regular inegral obained by subsiuing ς ε z insead of ς z in inegral I β of a3, ha is, I ε β = β ϕ ςdς ς ε z 2 nznς + ς ε z 2 nznς ϕ zdz. a5 27
S. G. Hong / Elecronic Journal of Boundary Elemens, Vol. 3, No., pp. -32 26 Then, by a4 we have I ε β = β ηl dη e e /L L η iε ξe 2 + ē ē /L L η + iε ξē 2 ξl dξ = β ηdη ξdξ η iε ξ 2 + η + iε ξ 2 = β + iε 2 log iε 2 + + iε 2 log iε 2 = β ε 2 logε 2 + ε 2 log + ε 2. The limi as ε is I β = β β logε 2, a6 where logε 2 is divergen erm. By he same way we can calculae I i β, i = 2, 3, 4. 2 e loge 2 + e e e I 2 β = β 2 + 2 + e + e e 2 + ē ē e logē + loge + e ē 2 + ē ē + ē ē loge logē ē logē + ē + β logε 2 which is sum of he finie inegral value and divergen erm β logε 2. I 3 β = β 2 e loge 2 + e loge 2 e e + 2 + e + e loge + e ē logē 2 + ē logē e e ē ē + 2 + ē + ē logē + ē + β logε 2, ē ē I 4 β = β β logε 2. 28
S. G. Hong / Elecronic Journal of Boundary Elemens, Vol. 3, No., pp. -32 26 Nex, we calculae I k β, k =, 2, 3, 4. I β = β ϕ ςdς = β nznς + nznς 2ς z ς z 2 + ς z 3 nznς ϕ zdz ηl dη + 2η ξe ē ē η ξē 3 L L = β ηdη η ξē 2 ξl dξ ē e + e ē L L L L e /ē + e /ē η ξ 2 2η ξe /ē η ξ 3 ξdξ =. I 2 β = β 2 e 3 e + e ē ē ē 2 e ē + ē 2 I 3 β = β 2 e + ē e ē + ē 2 ē + e logē e ē ē ē 2 e ē e ē + e ē ē 2 3 e e ē ē 2 ē + e e ē ē logē ē 2 e e + e ē ē ē ē 2 e logē ē logē + ē. e ē logē logē + ē, I 4 β =. A.4.2. calculaion of marix elemen D h β ϕ, ϕ, = Among inegrals in 43, I 2 β and I 2 β, being elemen-coinciden inegral, may conain divergen erms. Calculaing I 2 β by he boundary limi mehod, 29
S. G. Hong / Elecronic Journal of Boundary Elemens, Vol. 3, No., pp. -32 26 we ge I ε 2β = = β + ϕ + ςdς + ς ε z 2 nznς + ς ε z 2 nznς ϕ zdz = β ηdη η iε ξ 2 + η + iε ξ 2 ξdξ = β 2 log iε 2 + iε 2 log iε 2 2 + log iε 2 + iε 2 log iε 2 iε 2 log iε 2 + + iε 2 log iε 2. Here, log iε 2 = logε 2 is divergen erm and + iε 2 log iε 2 = + + ε 2 logε 2 + logε 2 ε is also divergen erm, bu hese divergen erms cancel each oher ou. Also, iε 2 log iε 2 = ε 2 logε 2 as ε, log iε 2 + iε 2 log iε 2 + + iε 2 log iε 2 = log + ε 2 ε 2 log + ε 2 + ε 2 log + ε 2, as ε. Therefore, we ge ha I 2 β = lim ε I ε 2β = β. I 2 β is elemen-coinciden inegral oo, bu have no divergen erm as former subsecion. Oher elemen inegrals are weak singular because heir inerior and exerior inegral inervals inersec only a end poin. We see I β only. I β = β = β = β 2 + + ē ηdη + 2 ϕ + ςdς nznς ς 2 + nznς ς z 2 ϕ zdz e e ηe + ξe 2 + ē ē ηē + ξē 2 ξdξ e e loge + e loge + e e e e ē ē ē η ηe log ηe + e logē + ē + e e η ē ηē log ηē loge + ē logē ē. η= ē ē logē 3
S. G. Hong / Elecronic Journal of Boundary Elemens, Vol. 3, No., pp. -32 26 Since η e ηe log ηe + η ē ηē log ηē =, we ge η= I β = β 2 e loge + e loge ē logē 2 e e ē + ē e logē + e loge + e ē e e ē + ē logē + ē. ē ē References [] Aimi, A., Diligeni, M., Hypersingular kernel inegraion in 3D Galerkin boundary elemen mehod, Journal of Compuaional and Applied Mahemaics 3822, 5-72. [2] Akinson, K., The Numerical Soluion of Inegral Equaions of he Second Kind, Cambridge Universiy Press, 997 [3] Bakar, S. A., Saleh, A. L., A echnique o remove second order singulariy applicaion in he boundary elemen mehod for elasoplasiciy plane sress analysis, Elecronic Journal of Boundary Elemens, 82, - 9. [4] Beylkin, G., Cramer, A., A muliresoluion approach o regularizaion of singular operaors and fas summaion, SIAM J. Sci. Compu., 2422, 8-7. [5] Cho, M. H., Cai, W., A wideband fas mulipole mehod for he wodimensional complex Helmholz equaion, Compuer Physics Communicaions, 8 2, 286-29. [6] Das, R., Hoa, M. K. and Bej, M., Some derivaive-free quadraure rules for numerical approximaions of Cauchy principal value of inegrals,hindawi Publishing Corporaion ISRN, Compuaional Mahemaics, Volume 24, Aricle ID 86397-8. [7] Eck, C., Seinbach, O. and Wendland, W. L., A symmeric boundary elemen mehod for conac problems wih fricion, Mahemaics and Compuers in Simulaion, 5999, 43-6. [8] Emmanuel, B., A complex parameer boundary elemen mehod for modeling AC impedances of elecrochemical sysems by analyic coninuaion- Applicaion o a sli geomery, Engineering Analysis wih Boundary Elemens, 3228, 77-777. 3
S. G. Hong / Elecronic Journal of Boundary Elemens, Vol. 3, No., pp. -32 26 [9] Fedoov, V. P., Spevak, L. F., One approach o he derivaion of exac inegraion formulae in he boundary elemen mehod, Engineering Analysis wih Boundary Elemens, 3228, 883-888. [] Grecu, L., Grecu, V., Demian, M. and Demian, G., A complex variable boundary elemen mehod for he flow around obsacles, Proceedings of he World Congress on Engineering 29, Vol II, WCE 29, July - 3, London, U.K., 29. [] Hsiao, G. C., Wendland, W. L., Boundary Inegral Equaions, Springer, 28. [2] Liu, Y., Chen, S., A new form of he hypersingular boundary inegral equaion for 3-D acousics and is implemenaion wih Cboundary elemens, Compu. Meh. Appl. Mech. Engrg., 73999, 375-386. [3] Liu, Y. J., Nishimura, N. and Yao, Z. H., A fas mulipole acceleraed mehod of fundamenal soluions for poenial problems, Engineering Analysis wih Boundary Elemens, 2925, 6-24. [4] Liu, Y. J., A new fas mulipole boundary elemen mehod for solving 2-D Sokes flow problems based on a dual BIE formulaion, Engineering Analysis wih Boundary Elemens, 3228, 39-5. [5] Liu, Y. J., A new fas mulipole boundary elemen mehod for solving large scale wo-dimenional elasosaic problems, Inernaional Journal for Numerical Mehods in Engineering, 65625, 863-88. [6] Nesvi, K. V., Scaering of TE Wave on Screened Pre-Canor Graing Based on Hypersingular Inegral Equaions,American Journal of Compuaional and Applied Mahemaics, 424, 9-6. [7] Of, G., Seinbach, O. and Wendland, W. L., The fas mulipole boundary elemen mehod for he symmeric boundary inegral formulaion, IABEM 22, Inernaional Associaion for Boundary Elemen Mehods, UT Ausin, TX, USA, May 28-3, 22. [8] Panzecaa, T., Cucco, F. and Terravecchia, S., Symmeric boundary elemen mehod versus finie elemen mehod, Compu. Mehods Appl. Mech. Engrg. 922, 3347-3367. [9] Suradhar, A., Paulino, G. H. and Gray, L. T., Symmeric Galerkin boundary elemen mehod, Springer, 28. [2] Sao, K., Complex variable boundary elemen mehod for poenial flow wih hin objecs, Compu. Mehods Appl. Mech. Engrg., 9223, 42-433. [2] Vladimirescu, I., Grecu, L., Weakening he Singulariies when applying he BEM for 2D compressible fluid flow, Proccedings of he Inernaional Muli- Conferece of Engineers and Compuer Scieniss 2: Vol III,IMECS 2, March 7-9, Hong Kong, 2. 32