A Difference-Analytical Method for Solving Laplace s Boundary Value Problem with Singularities

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1 5 10 July 2004, Antalya, urkey Dynamical Systems and Applications, Proceedings, pp A Difference-Analytical Metod for Solving Laplace s Boundary Value Problem wit Singularities A. A. Dosiyev and S. Cival Department of Matematics, Eastern Mediterranean University, Gazimagusa, Cyprus, Mersin 10, urkey adiguzel.dosiyev@emu.edu.tr suzan.buranay@emu.edu.tr Abstract Adifference-analytical metod of solving te mixed boundary value problem for Laplace s equation on polygons (wic can ave broken sections and be multiply connected) is described and ustified. e uniform estimate for te error of te approximate solution is of order O 2,were is te mes step, for te errors of te derivatives of order p, p =1, 2,..., in a finite neigbourood of vertices, of order O( 2 /r p 1/λ ),werer is te distance from te current point to te vertex in question, λ =1/α or λ =1/(2α ),dependingonte types of boundary conditions, α π is te value of te angle. e last part of te paper is devoted to illustrate numerical experiments. 1 Introduction It is well known tat te use of classical finite difference or finite element metods to solve te elliptic boundary value problems wit singularities becomes ineffective. A special construction is usually needed for te numerical sceme near te singularities in suc a way tat te order of convergence is te same as in te case of a smoot solution. Among many approaces to solve tis problem, a special empasis as been placed on te construction of combined metods, in wic differential properties of te solution in different parts of te domain are used, and an effective realization of te obtained system of algebraic equations is acieved (see [1], [2] [5], and references terein). In [2] [4] a new combined difference-analytical metod called te Block-Grid Metod in solving te Laplace equation on graduated polygons is introduced. is metod is a combination of two metods wic takes only superiorities of eac 339

2 340 A. A. Dosiyev and S. Cival one of tem: te exponentially convergent Block Metod (see [6, 7] and references terein) wic finely takes into account te beavior of te exact solution near te vertices of interior angles 6= π/2 of polygon (on singular part) and te Finite Difference Metod, wic as a simple structure and ig accuracy on square grids of te rectangles covering te remainder, nonsingular part of te polygon. A sixt order gluing operator S 6 is constructed for gluing togeter te grids and blocks. e expression S 6 u P ξ k u k contains te function values at 31 grid nodes. e pattern of te gluing operator S 6 makes restriction of te polygon as to be graduated. In tis paper, we use sufficiently simple linear interpolation as a gluing operator wic contains te function values at fourt grids, and te Block-Grid Metod is extended for te mixed boundary value problems on arbitrary polygons. e uniform estimate for te error of te approximate solution is of order O( 2 ), and for te errors of te derivatives of order p, p =1, 2,..., in te blocks is O( 2 /r p λ ), were r is te distance from te current point to te vertex in question, and λ depends on te magnitude of te interior angle and te type of boundary conditions on te sides of te angle considered. Furtermore, wen te boundary conditions are eiter Diriclet or mixed type on te sides of te interior angles 6= π/2, te error of te approximate solution on te block sectors decreases as r λ 2, wic gives te additional accuracy of tis approac near te singular points, wit respect to existing finite difference or finite element modifications for te singular problems. e system of finite difference equations on te union of all rectangles may be solved by te alternating metod of Scwarz wit te number of iterations O(ln ε 1 ), were ε is te prescribed accuracy, by solving standard 5-point difference equations of Laplace on rectangular domain at eac iteration. Finally, we illustrate te effectiveness of te metod in solving te problem in an L-saped polygon wit te corner and boundary singularities, and te well known Motz problem. 2 Boundary Value Problem on Polygons Let G be an open simply connected polygon, γ, = 1,N, be its sides, including te endpoints, enumerated counterclockwise, γ = γ 1 γ N be te boundary of G, α π, 0 <α 2, be te interior angle formed by te sides γ 1 and γ (γ 0 = γ N ), A = γ 1 γ be te vertex of te t angle, r,θ be a polar system of coordinates wit te pole at A and te angle θ taken counterclockwise from te side γ,ν be a parameter taking te values 0 or 1, andν =1 ν. We consider te boundary value problem u = 0 on G, (1) ν u + ν u (1) n = ν ϕ + ν ψ on γ,= 1,N, (2)

3 A difference-analytical metod for solving Laplace s BVP wit singularities 341 were 2 / x / y 2,u (1) n is te derivative along te inner normal, ϕ and ψ are given functions of te arc lengt s taken along γ and 1 ν 1 + ν ν N N, (3) ν ϕ + ν ψ C 2,λ (γ ), 0 <λ<1, 1 N, (4) at te vertices A (s = s ) for α =1/2 te conugation conditions ϕ 1 = ϕ for ν 1 = ν =1; ϕ 0 1 = ψ for ν 1 = ν =0; ψ 1 = ϕ 0 for ν 1 = ν =0; ψ 0 1 = ψ 0 for ν 1 = ν =0, (5) were ϕ 0 µ,ψ 0 υ are te derivatives along te boundary arc, are satisfied. At te vertices A wit α 6= 1/2 no compatibility conditions are required to old for te boundary functions, in particular, te values of ϕ 1 and ϕ at A migt be different. In addition, we require tat, wen α 6=1/2, te boundary functions on γ 1 and on γ be given as algebraic polynomials of s. We represent te given boundary functions (algebraic polynomials) on γ 1 and γ for α 6=1/2in te form τ 1 τ and b 0 k rk, (6) k=0 a 0 k rk respectively, were a 0 k and b0 k are numerical coefficients and τ 1 and τ are te degrees of tose polynomials. Let E be te set (1 N) for wic α 6=1/2. In te neigborood of A, E, weconstructtwofixed block-sectors i = (r i ) G, i =1, 2, were 0 <r 2 <r 1 < min{s +1 s,s s 1 }, (r) ={(r,θ ):0<r <r, 0 <θ <α π}. On te closed sector 1, E, we consider a function Q (r,θ ) wit te following properties: (a) Q (r,θ ) is armonic and bounded on te open sector 1; (b) continuous on 1 everywere, except for te point A (te vertex of te sector) for ν = ν 1 =1and a 0 0 6= b0 0, i.e., for discontinuous at A given boundary values; (c) continuously differentiable on 1 \ A and satisfies te boundary conditions (2) on γ 1 1 and γ 1, E. For definiteness we assume tat Q (r,θ ) wit te above properties as te form (3.2) (3.9) given in [7]. Remark 1 For te case of ν 1 = ν =1we formally set te value of Q (r,θ ) and te solution u of problem (1), (2) at te vertex A equal to (a b0 0 )/2. k=0

4 342 A. A. Dosiyev and S. Cival We set (see [6]) R(m, m, r, θ, η) = R(r, θ, η)+( 1) m R(r, θ, η), (7) R(1 m, m, r, θ, η) = R(m, m, r, θ, η) ( 1) m R(m, m, r, θ, π η), (8) were 1 r 2 R(r, θ, η) = 2π(1 2r cos(θ η)+r 2 ), (9) is te kernel of te Poisson integral for a unit circle. We specify te kernel µ λ r R (r,θ,η)=λ R(ν 1, ν,,λ θ,λ η), E, (10) r 2 were λ = 1 (2 ν 1 ν ν 1 ν )α (11) Lemma 2 (Volkov [7]) e solution u of te boundary value problem (1), (2) can be represented on 2 \ V, E, in te form Z α π u(r,θ )=Q (r,θ )+ R (r,θ,η)(u(r 2,η) Q (r 2,η)) dη, (12) 0 were V is te curvilinear part of te boundary of 2. 3 Description of te Block-Grid Metod Let us consider in addition to te sectors 1 and 2 (see 2) in te neigborood of eac vertex A, E, of te polygon G te sectors 3 and 4,were0 < r 4 <r 3 <r 2, r 3 =(r 2 + r 4 )/2 and k 3 l 3 =, k 6= l, k, l E, and let G = G \ ( E 4). Let Π k G,k= 1,M (M < ) be certain fixed open rectangles wit arbitrary orientation, generally speaking, wit sides a 1k and a 2k,a 1k /a 2k being rational and G =( M k=1 Π k) ( E 3) (see Figures 1 and 3 in Section 6). Let η k be te boundary of te rectangle Π k and V be te curvilinear part of te boundary of te sector 2,andt =( M k=1 η k) 3. e following general requirement is imposed on te arrangement of te rectangles Π k, k = 1,M : any point P lying on η k G, 1 k M, or located on V G, E, falls inside at least one of te rectangles Π k(p), 1 k(p) M, depending on P, and te distance from P to G ηk(p) is not less tan some constant κ 0 > 0 independent of P.

5 A difference-analytical metod for solving Laplace s BVP wit singularities 343 We call te quantity κ 0 a dept of gluing of te rectangles Π k,k= 1,M.We introduce te parameter (0, κ 0 /2] and define a square grid on Π k,k= 1,M, wit maximal possible step k min{, min{a 1k,a 2k }/2} suc tat te boundary η k lies entirely on te grid lines. Let Π k be te set of grid nodes on Π k,andη k be te set of nodes on η k ; Π k = Π k η k,η k0 is te set of nodes on te closure of η k G,t is te set of nodes on t,andη k1 is te set of remaining nodes on η k. We also introduce te natural number n and te quantities =max{4, [α n]}, β = α π/, and θ m =(m 1/2)β, E, 1 m. On te arc V we coose te points (r 2,θ m ), 1 m, and denote te set of tese points by V n. Let ω,n =( M k=1 η k0 ) ( EV n ), G,n = ω,n ( M k=1 Π k). Let R (q) (r,θ )= R (r,θ,θ q n ) P o, (13) max 1,β p=1 R (r,θ,θ p ) were R (r, θ, η) is te kernel defined by (10). It is easy to ceck tat 0 R (q) (r,θ ) R (r,θ,θ q ), (14) were E, 0 q. Furtermore, from te estimation (2.29) in [6] tere follows te existence of te positive constants n 0 and σ>0 suc tat for n n 0, max 3 β wen ν 1 + ν 1, and on te basis of (13) and (14) R (r,θ,θ q ) σ<1, (15) 0 β R (q) (r,θ ) 1, E, (16) wen ν 1 = ν =0. We define operators S 2,A,L and <, E, on te sets ω,n, Π k,η k1 and t, respectively. e operator S 2 is called a gluing operator, wic is defined at eac point P ω,n in te following way. We consider te set of all rectangles {Π k } in te intersections of wic te point P lies, and we coose one of tese rectangles Π k(p ) part of wose boundary, situated in G is te furtest away from P. e value S 2 u at te point P is computed according to te values of te function at te four vertices P k,k=1, 2, 3, 4, of te closure of te cell, containing te point P,of

6 344 A. A. Dosiyev and S. Cival te grid constructed on Π k(p ), by multilinear interpolation in te directions of te grid lines. us, S 2 u as te expression were u = u(p ), u µ = u(p µ ), and S 2 u 4 λ µ u µ, (17) µ=1 λ µ 0, 4 λ µ =1. (18) µ=1 We define on Π k, 1 k M, te operator A of calculating te aritmetic mean of te function at te four neigboring points of te same net. Let L be te operator defined on te points η k1 as follows L(u, ϕ, ψ) = ϕ m (P ), P η k1 γ m, ν m =1; (19) L(u, ϕ, ψ) (u 1 + u 2 +2u 3 )/4 k ψ m (P )/2, (20) wen P η k1 γ m\(a m A m+1 ), ν m =0, were P 1 and P 2 beingtetwopoints of Π k on γ m closest to P,andP 3 being te point on Π k closest to P ; L(u, ϕ, ψ) (u 1 + u 2 k (ψ 1 (P )+ψ (P )))/2, ν 1 = ν =0, (21) P = A η k1,p 1 and P 2 being te two points in η k1 closest to P. Let us define te operator < on t, E, asfollows: < (u, ϕ, ψ) Q (r,θ )+β R (q) (r,θ )(u(r 2,θ q ) Q (r 2,θ q )), (22) were Q is te function definedinsection2. Consider te system of linear algebraic equation were 1 k M, E. u = Au on Π k, (23) u = L(u,ϕ,ψ) on η k1, (24) u (r,θ ) = <(u,ϕ,ψ) on t, (25) u = S 2 u on ω, (26)

7 A difference-analytical metod for solving Laplace s BVP wit singularities 345 Definition 3 e solution of te system (23) (26) is called a numerical solution of te problem (1), (2) on G,n. Definition 4 We consider te sector = (r ), were r =(r 2 + r 3 )/2, E. Let u be te solution of te system (23) (26). e function U (r,θ )=Q (r,θ )+β R (r,θ,θ q )(u (r 2,θ q ) Q (r 2,θ q )), (27) defined on, is called an approximate solution of te problem (1), (2) on te closed block 3, E. 4 Analysis of te system of block-grid equations 4.1 Solvability of system (23) (26) eorem 5 ere is a natural number n 0 suc tat, for all n n 0 te system (23) (26) as a unique solution. Proof. We consider a omogeneous system of finite difference equations corresponding to system (23) (26) v = Av on Π k,v = L(v, 0, 0) on η k1 γ m, v (r,θ ) = <(v, 0, 0) on t,v = S 2 v on ω,n, (28) 1 m N, 1 k M, E. Let te system (28) ave a solution v 6=0on G,n, and v = v (P )=max,n G v 6= 0. If P = P (r,θ ) t, ten from (22) and (28) we ave v (r,θ )=β n( ) R (q) (r,θ )S2 v (r 2,θ q ), (29) i.e., tevaluev (P )=v is linearly represented troug te values of te function v at te points of rectangular grids Π l(p ),l(p )=1, 2,...,M 0,M 0 M. Hence, wen ν 1 + ν 1, on te basis of (14), (15), (17), (18), and (29) it follows tat for n n 0 te function v cannot take te value v = v at te nodes t. Wen ν 1 + ν =0, from te gluing condition it follows tat, on te rigt-and side of (29) only an interior points of Π l, l =1, 2,..., M0, M 0 M, are used and by virtue of (16) (18) at tese points v = v. Similarly, if v (P )=v and

8 346 A. A. Dosiyev and S. Cival P = η k1 γ m,ν m =0, ten from te constructions of te operator L by (20) and (21) te function v takes te value v at some interior points of some rectangular grid. us, we can assume tat v (P )=v 6=0, P Π q for some fixed q, 1 q M. But on te basis of (3), (17), (18), (28), and te principle of maximum, tere exists a rectangular grid Π µ 0 wit te boundary η µ 0 1 γ m 6=,ν m =1,at wic v = v =0. is contradicts te assumption v 6= Error estimates on G,n Let ε = u u, (30) were u is a solution of system (23) (26), and u is te trace on G,n of te solution of (1), (2). On te basis of (1), (2), (23) (26) and (30) te error ε satisfies te system of difference equations ε = Aε + r 1 on Π k, ε = L(ε, 0, 0) + r 2 on η k1, ε (r,θ ) = β R (q) (r,θ )ε (r 2,θ q )+r3, (r,θ ) t, (31) were 1 k M, E, ε = S 2 ε + r 4 on ω,n, r 1 = Au u on M k=1 Π k, (32) r 2 = L(u, ϕ, ψ) u on M k=1 η k1, (33) r 3 = β R (q) (r,θ )(u(r 2,θ q ) Q (r 2,θ q )) (u(r,θ ) Q (r,θ )) on E t, (34) r 4 = S 2 u u on ω,n. (35) In wat follows and for simplicity, we will denote constants wic are independent of by c. Lemma 6 ere exists a natural number n 0 suc tat, for all ln 1+κ 1 +1}, were κ > 0 is a fixed number, n max{n 0, max r3 E c 2. (36)

9 A difference-analytical metod for solving Laplace s BVP wit singularities 347 Proof. On te basis of (34), Lemma 2 and by virtue of r 3 =(r 2 +r 4 )/2 <r 2 we ave r3 β Z α π 0 +β R (q) R (r,θ,θ q )(u(r 2,θ q ) Q (r 2,θ q )) R (r,θ,η)(u(r 2,η) Q (r 2,η)) dη (r,θ ) R (r,θ,θ q ) u(r 2,θ q ) Q (r 2,θ q ). From tis and from te Lemmas 2.5 and 2.10 given in [6] and taking te boundedness of te difference u(r 2,θ q ) Q (r 2,θ q ), 1 q, into account, we obtain c 0 exp d 0 n ª, E, (37) r 3 o were c 0 and d0 > 0 are, independent of n, constants. Putting c0 =max E nc 0, and d =min{d },from(37)weave max r 3 c 0 exp d 0 n ª. (38) E Let n 0 be a natural number suc tat te inequality (15) olds. en, for all n max{n 0, ln 1+κ 1 +1}, were κ > 0 is a fixed number, we ave te inequality (36). Since te set of points ω,n, according to te construction, is located from te vertices of te polygon G at a distance exceeding some positive quantity independent of, ten by virtue of (4), estimation (4.64) obtained in [8], from (35) we ave max r 4 ω,n c 2. (39) eorem 7 Assume tat conditions (3) (5) old. en, tere exists a natural number n 0 suc tat, for all n max n 0, ln 1+κ 1 +1 ª, were κ > 0 is a fixed number, max u u c 2. G,n Proof. Let us take an arbitrary rectangular grid Π k and let t k = Π k t. Let t k 6=, and v be a solution of system (31) in te case wen te discrepancies

10 348 A. A. Dosiyev and S. Cival r 1,r2,r3, and r4 in Π k are te same as in (32) (35), but are zero in G,n \ Π k. By analogy to te proof of eorem 5 one can sow tat W =max v =max v. (40) G,n Π k We represent te function v on G,n in te form v = 4 v κ, (41) κ=1 were te functions v κ,κ=2, 3, 4, aredefined on Π k of equations as a solution of te system v 2 = Av 2 on Π k,v2 = L(v2, 0, 0) on η k 1, v 2 (r,θ ) = r 3, (r,θ ) t k, v2 =0 on ω,n ; (42) v 3 = Av 3 on Π k,v3 = L(v3, 0, 0) on η k 1, v 3 (r,θ ) = 0, (r,θ ) t k, v3 = r4 on ω,n ; (43) v 4 = Av 4 + r1 on Π k,v4 = L(v4, 0, 0) + r2 on η k 1, v 4 (r,θ ) = 0, (r,θ ) t k, v4 =0 on ω,n, (44) wit v κ =0,κ=2, 3, 4, on G,n \Π k. (45) Hence according to (41) (45) te function v 1 satisfies te system of equations v 1 = Av 1 on Π k, v1 = L(v1, 0, 0) on η k1, v 1 (r 4,θ ) = β R (q) (r,θ ) v κ (r 2,θ q ), (r,θ ) t, (46) v 1 = S 2 Ã 4 κ=1 v κ! κ=1 on η k0, 1 k M, E, were te functions v κ,κ=2, 3, 4 are assumed to be known. aking into account (36) and (39), on te basis of (42), (43), (45) and te maximum principle we ave W 2 = maxv2 c 2, (47) G,n W 3 = maxv3 c 2. (48) G,n

11 A difference-analytical metod for solving Laplace s BVP wit singularities 349 e function v 4 to be a solution of te system (44) wit (45) is te error of te finite difference solution, wit step k, of te boundary value problem for Laplace s equation on Π k. en, by virtue of (45) and eorem 3.1 in [9], we obtain W 4 =maxv 4 G,n =max v 4 Π c 2. (49) k We estimate te function v 1, wic, according to eorem 5, is te unique solution of system (46). On te basis of (15) (18) and te gluing condition of te rectangles Π k,k= 1, 2,...,M, from (46) by means of [9], tere exists a real number λ, 0 <λ < 1, independent of, suc tat for all n max n 0, ln 1+κ 1 +1 ª we ave W 1 =max G,n v 1 From (40), (41), (47) (50) we obtain W = λ W +2 λ W + 4 i=2 max G,n v i 4 W i λ W + c 2, 0 <λ < 1, i=2. (50) i.e., W =max v c 2. (51) G,n Intecasewent k, te function v 2 0 on G,n and te inequality (51) olds true. Since te number of grid rectangles in G,n is finite, for te solution of (31) we ave max ε c 2. G,n 4.3 Convergence of te approximate solution on blocks We consider te question of convergence of te function U (r,θ ) defined by te formula (27). aking into account te properties of te functions Q (r,θ ), E, andtefacttattekernelr (r,θ,η) satisfies te omogenous boundary condition definedby(2)on(γ 1 γ ) 2, te function U (r,θ ) is bounded, armonic on and continuous up to its boundary, except for te vertex A wen te specified boundary values are discontinuous at A. In addition, on te rectilinear parts of te boundary of, except, maybe, te vertex A, te function U (r,θ ) satisfies te boundary conditions definedin(2).

12 350 A. A. Dosiyev and S. Cival eorem 8 ere is a natural number n 0, suc tat for all n max{n 0, ln 1+κ 1 +1}, κ > 0 is a fixed number, te following inequalities are valid: p x p q y q (U (r,θ ) u(r,θ )) c p 2 on 3, (52) first, for integer λ and any ν 1 and ν wen p λ, second, for ν 1 = ν =0 and any λ wen p =0; p x p q y q (U (r,θ ) u(r,θ )) c p 2 /r p λ on 3, (53) for any λ, if ν 1 + ν 1, 0 p<λ or ν 1 = ν =0, 1 p<λ ; p x p q y q (U (r,θ ) u(r,θ )) c p 2 /r p λ on 3\A, (54) for noninteger λ, and any ν 1 and ν wen p>λ. Everywere 0 q p, λ is te quantity (11), ν 1 and ν are parameters entering into te boundary conditions (2), u is a solution of te problem (1), (2). Proof. On te bases of (27) and Lemma 2, on te closed block, E, we ave U (r,θ ) u(r,θ ) = β R (r,θ,θ q )(u(r 2,θ q ) Q (r 2,θ q )) Z α π 0 +β R (r,θ,η)(u(r 2,η) Q (r 2,η)) dη R (r,θ,θ q )(u (r 2,θ q ) u(r 2,θ q )). (55) Since r =(r 2+r 3 )/r 2, by analogy of te proof of Lemma 6 for n ln 1+κ 1 + 1, κ > 0 is a fixed number, we ave β Z α π 0 R (r,θ,θ q )(u(r 2,θ q ) Q (r,θ q )) R (r,θ,η)(u(r 2,η) Q (r 2,η)) dη c 2, on, E. (56)

13 A difference-analytical metod for solving Laplace s BVP wit singularities 351 On te basis of (15) and eorem 7 for all n max n 0, ln 1+κ 1 +1 ª we obtain R (r,θ,θ β q )(u (r 2,θ q ) u(r 2,θ q )) c2, on, E. (57) From (55) (57) for all n max n 0, ln 1+κ 1 +1 ª we ave U (r,θ ) u(r,θ ) c 2 on, E. (58) Let ε (r,θ )=U (r,θ ) u(r,θ ) on, E. (59) From (27), (59), and Remark 1 it follows tat te function ε (r,θ ) is continuous on, and is a solution of te boundary value problem ε = 0 on, ν m ε + ν m (ε ) 0 n = 0 on γ m,m= 1,, (60) ε (r,θ ) = U (r,θ ) u(r,θ ), 0 θ α π. Since 3, E, taking into account (58)-(60), from te Lemma 6.12 in [7] follows all inequalities of eorem 8. 5 e use of Scwarz s alternating metod to solve te system of block-grid equations According to Definitions 3 and 4, te approximate solution of problem (1), (2) must first be found in te domain G,n as te solution of te system of difference equations (23)-(26), and te solution itself and its derivatives of order p, p =1, 2,..., at any point of 3, E, except may be te vertex A, can ten be found using formula (21). erefore, it is sufficient to ustify te possibility of finding a solution of system (23) (26) by Scwarz s alternating metod. We denote by γ D te union of all sides of te polygon G on wic te boundary condition is of Diriclet type. From (3) it follows tat γ D 6=. We define te following classes Φ τ, τ =1, 2,...,τ, of rectangles Π k, k =1, 2,...,M (see [3]). e class Φ 1 includes all rectangles wose intersection wit γ D contains a certain segment of positive lengt. e class Φ 2 contains all te rectangles wic are not in te class Φ 1, wose intersection wit rectangles of Φ 1 contains a segment of finite

14 352 A. A. Dosiyev and S. Cival lengt, and so on. Let Π k0 be te set of nodes of te grid Π k wic are not less tan l 0 =min{min 1 k M min {a 1k,a 2k },ν 0 } /4 from te set η k0. Let Φ τ0 = [ k:π k Φ τ Π k0,τ =1, 2,...,τ ; G 0 = τ [ τ=1 Φ τ0. We use Scwarz s alternating metod to solve system (23) (26) in te following form. Suppose we are given a zero approximation u (0) to te exact solution u of (23) (26). Finding u (1) for all E wit te formula (25) on t and wit (26) on η k0, we solve system (23) (26) on grids Π k constructed on rectangles belonging to te class Φ 1, andtenclassφ 2 and so on. e next iteration is similar. Consequently, we ave te sequence u (1),u(2),..., defined as follows u (m) (r,θ ) = Q (r,θ ) +β R (q) (r,θ )(u (m 1) (r 2,θ q ) Q (r 2,θ q )) on t, u (m) = S 2 u (m 1) on ω,n, (61) u (m) = Au (m) on Π k, u(m) = L( u (m),ϕ,ψ) on η k1, were 1 k M, E, m =1, 2,.... eorem 9 For any n max n 0, ln 1+κ 1 +1 ª te system (23) (26) can be solved by Scwarz s alternating metod wit any accuracy ε>0 in a uniform metric wit te number of iterations O(ln ε 1 ), independent of and n, were n 0 and κ mean te same as in eorem 8. Proof. Let ε (m) = u (m) u on G,n, (62) were u is te exact solution of (23) (26), and u (m) is te m t iteration defined by (61), m =1, 2,... en for any m we ave ε (m) (r,θ ) = β R (q) (r,θ )S 2 (ε (m 1) (r 2,θ q )) on t η k, (63) ε (m) = S 2 (ε (m 1) ) on η k0, (64) ε (m) = Aε (m) on Π k, ε(m) = L( ε (m), 0, 0) on η k1, (65)

15 A difference-analytical metod for solving Laplace s BVP wit singularities 353 were 1 k M, E. We denote W (m) =max G,n ε (m) From (15) (18), (63), and (64) for n n 0 we ave max ε (m) M max ε (m 1) k=1 η k,0 G W (m 1), (66) 0 max ε (m) E t (r,θ ) max ε (m 1) W (m 1),m=1, 2,... (67) G 0 By virtue of (66), (67) and te maximum principle, from (65) we obtain W (0). W (1) W (2), (68) max ε (m 1),m=1, 2,... (69) W (m) G 0 aking into account te sequence of calculations over te classes Φ τ, τ = 1, 2,...,τ, and te inequalities (68) and (69), by means of [3] we obtain W (m+1) µ m (0) τ W, m =1, 2,..., (70) were µ τ < 1 is independent of m and. From (70) tere follows te statement of eorem 9. Remark 10 If on te sides of te rigt interior angles of te polygon G te boundary functions are given also as algebraic polynomials of s, ten, witout te conugation conditions (5), te approximate solution in neigboroods of te vertices of te rigt angles can be defined by te formula (27), and any order derivatives can be found by its simple differentiation. Remark 11 From te error estimation formula (53) of eorem 8 it follows tat, wen on te sides of te interior angles te boundary conditions are eiter Diriclet or mixed type, te error of te approximate solution on te block sectors decreases as r λ 2, wic gives an additional accuracy of te Block Grid metod near te singular points, wit respect to existing finite difference or finite element modifications for te singular problems. Remark 12 e metod and results carry out witout canges to multiply connected polygons.

16 354 A. A. Dosiyev and S. Cival 6 Numerical examples o test te effectiveness of te Block-Grid metod, we computed two numerical examples. In Example 13, te polygon G is L-saped (Fig. 1 ), and te exact solution as bot te boundary (discontinuity of te boundary functions) and te angle singularities at te vertex A 1 of te interior angle (α 1 π =3π/2). In Example 14, te exact solution as singularities at te vertex A 1, wit te interior angle α 1 π = π, caused by abrupt canges in te type of boundary conditions. In Example 13 te comparisons are made between te exact and te approximate solutions, and teir derivatives. In Example 14, because of boundary conditions on te sides for y 0 (see Fig. 3), te exact solution is unknown and comparisons are made at various points wit values from te literature. In Example 13, te four overlapping rectangles Π k, k =1,...,4, andinexample 14 tree overlapping rectangles Π k, k =1, 2, 3, are taken as sown on Figs. 1 and 3 respectively. We found te solution in eac rectangle on te mes segments η k0, k =1,...,6, in Example 13 and k =1,...,4, in Example 14. After obtaining te prescribed accuracy ε = on te boundary of eac standard rectangle, te system on eac rectangle in (61) is solved by te formulas in [10], and [11], wit te use of discrete fast Fourier transform. Furtermore, according to te boundary conditions on γ 0 and γ 1 for Example 13, te armonic function Q 1 (r, θ) =θ, and for Example 14, Q 1 (r, θ) =0, and in bot examples te radius r 1,2 of sector 1 2 are taken In all tables below, U is used as te numerical solution obtained by Block-Grid metod and u is te exact solution of te given problem. Example 13 Let G be L-saped, and defined as follows (see Fig. 1) G = {(x, y) : 1 <x<1, 1 <y<1}\g 1, were G 1 = {(x, y) : 0 x 1, 1 y 0}. Let γ be te boundary of G. Consider te following problem: u = 0 on G, (71) u = v(r, θ) on γ, (72) were v(r, θ) =θ + r 2 3 sin 2θ 3 is te exact solution of tis problem. In all tables te following notations are used: Π 1 = G\( M k=1 Π k), G NS = G\Π 1 nonsingular part of G; G S = G Π 1 singular part of G, G S = G S {r }, G,n NS = G NS G,n, kwk C(Ω) =max Ω w. In ables 1 3 te errors ε = U u, ε (1,0) ³ r 1/3 U y u y ³ = r 1/3 U x u x ³,ε (2,0) = r 4/3 2 U x 2 2 u ³ x 2,ε (1,1) = r 4/3 2 U x y 2 u x y, ε (0,1) = in te

17 A difference-analytical metod for solving Laplace s BVP wit singularities Π Π 1 * Π 4 0 A Π Π Figure 1: L-saped domain maximum norm between te block-grid solution U and te exact solution u of te problem in Example 13 are given. ( 1,n) kε k C(G,n NS ) kε k C(GS ) kε k C(G 0.25 S ) kε k C(G S ) Iter. (8, 20) 1.17D D D D 5 13 (8, 30) 9.39D D D D 6 13 (8, 40) 9.17D D D D 6 13 (16, 60) 2.43D D D D 7 13 (32, 40) 6.23D D D D 6 14 (32, 50) 6.13D D D D 7 14 (32, 60) 5.93D D D D 7 14 (64, 50) 1.53D D D D 7 15 able 1: kε k for L-saped problem and r 12 =0.93

18 356 A. A. Dosiyev and S. Cival ( 1,n) kε (1,0) k C(GS ) kε (1,0) k C(G 0.25 S ) kε (1,0) (8, 30) 2.81D D D 6 (8, 40) 7.29D D D 5 (8, 50) 6.46D D D 6 (16, 60) 1.81D D D 6 (32, 40) 1.89D D D 5 (32, 60) 3.68D D D 6 (64, 60) 2.31D D D 7 (64, 70) 8.32D D D 6 k C(G S ) able 2: kε (1,0) k for L-saped problem and r 12 =0.93 ( 1,n) kε (2,0) k C(GS ) kε (2,0) k C(G 0.25 S ) kε (2,0) (8, 30) 2.28D D D 7 (8, 40) 8.18D D D 6 (8, 50) 5.83D D D 6 (16, 60) 1.46D D D 6 (32, 60) 2.70D D D 6 (64, 50) 1.93D D D 6 k C(G S ) able 3: kε (2,0) k for L-saped problem and r 12 =0.93 Example 14 (Motz problem [15]). Let G = {(x, y) : 1 <x<1, 0 <y<1}, and γ be its boundary (Fig. 3). We consider te following problem: u = 0 in G, (73) u = 0 on y =0, 1 x 0, u = 500 on x =1, u n = 0 on te oter boundary segments of γ. e exact solution of tis problem is unknown. In able 4 te solution of te Motz problem at various points obtained by block-grid metod wen ( 1 =32,n= 60), and ( 1 =64,n = 80) is compared wit te values from [12]. e results of [12] are presented after a correction of te 31-t coefficient (divided by 10), in te expansion of te approximate solution, discovered in [13], and are called an extremely accurate solution (see also [14]).

19 A difference-analytical metod for solving Laplace s BVP wit singularities L-saped problem r 12 =0.93, =1/ L-saped problem r 12 =0.93, =1/16 maxerror maxerror L-saped problem n r =0.93, =1/ L-saped problem n r =0.93, =1/ maxerror maxerror ε (GNS ) ε (GS ) ε 0.5 (GS ) ε 0.25 (GS ) n n ε (GS ) Figure 2: Decreasing error around te singular point (x i,y i ) 1 =32,n=60 1 =64,n=80 Li et al.[1987] ( 2 7, 2 7 ) (0, 2 7 ) ( 2 7, 2 7 ) (0, 1 7 ) ( 1 28, 1 28 ) (0, 1 28 ) ( 1 28, 1 28 ) ( 28 1, 0) ( 3 28, 0) , 0) ( 1 7 able 4: Solution of te Motz problem at various points compared wit values from literature

20 358 A. A. Dosiyev and S. Cival u/ n =0 on γ u/ n =0 on γ Π 2 u=500 on γ Π 1 Π 1 * Π 3 0 u=0 on γ A 0 1 u/ n =0 on γ 1 r 12 r Concluding remarks Figure 3: Motz problem In te proposed metod, te given polygon is decomposed into a finite number of overlapping rectangles (nonsingular parts) and sectors (singular parts). In te sectors we approximate te special integral representation of te solution wic finely takes te beavior of te exact solution near te vertices of te interior angles of te polygon into account and on rectangles te 5-point sceme is used to approximate Laplace s equation on square grids wic is simpler by means of sparsity. A gluing togeter of te subsystems is carried out, effected by a sufficiently simple linear interpolation. ese properties are a prerequisite for a ig rate of convergence establised by eorems 7 and 8, and an effective realization given in Sections 5 and 6. Furtermore, on te singular parts any order derivatives of te solution at any point can be approximated by simple differentiation of te function (27) (see eorems 8 and ables 2, 3. is issue seems to be more problematic for finite difference, finite element, and teir existing modifications.

21 A difference-analytical metod for solving Laplace s BVP wit singularities 359 Acknowledgment e autors tanks Professor E. A. Volkov for is attention to tis work and for valuable advice. References [1] Li Z. C., Combined Metods for Elliptic Problems wit Singularities, Interfaces and Infinities, Kluwer Academic Publisers, Dordrec, Boston and London, [2] Dosiyev A. A., A block-grid metod for increasing accuracy in te solution of te Laplace equation on polygons, Russian Acad. Sci. Dokl.Mat., 45 (1992), No. 2, [3] Dosiyev A. A., A block-grid metod of increased accuracy for solving Diriclet s problem for Laplace s equation on polygons, Comp. Mats Mat. Pys., 34 (1994), No. 5, [4] Dosiyev A. A., eigaccurateblock-gridmetodforsolvinglaplace sboundary value problem wit singularities, SIAM Journal on Numerical Analysis, 42 (2004), No. 1, [5] Dosiyev A. A., A fourt order accurate composite grids metod for solving Laplace s boundary value problems wit singularities, Comp. Mats Mat. Pys., 42 (2002), No. 5, [6] Volkov E. A., An exponentially converging metod for solving Laplace s equation on polygons, Mat. USSR Sb., 37 (1980), No. 3, [7] Volkov E. A., Block Metod for Solving te Laplace Equation and Constructing Conformal Mappings, USA, CRC Press, [8] Volkov E. A., On differentiability properties of solutions of boundary value problems for te Laplace s equation on polygons, r. Mat. Inst. Akad. Nauk SSSR, 77 (1965), [9] Volkov E. A., On te metod of composite meses for Laplace s equation on polygons, rudymat. Inst. Steklov, 140 (1976), [10] Wasow W., On te truncation error in te solution of Laplace s equation by finite differences, J. Res. Nat. Bur. Standards, 48 (1952), [11] Romanova S. E., An efficient metod for te approximate solution of te Laplace difference equation on rectangular domains, Z. Vycisl. Mat. i Mat. Fiz., 23 (1983),

22 360 A. A. Dosiyev and S. Cival [12] Li Z. C., Maton R. and Sermer P., Boundary metods for solving elliptic problems wit singularities and interfaces, SIAMJ. Numer. Anal., 24 (1987), No. 3, [13] Lucas.R.andOH.S., e metod of auxiliary mapping for te finite element solutions of elliptic problems containing singularities, J. Comput. Pys., 108 (1993), [14] Wu. and Han H., A finite-element metod for Laplace- and Helmoltz-type boundary value problems wit singularities, SIAMJ. Numer. Anal., 34 (1997), No. 3, [15] Motz H., e treatment of singularities of partial differential equations by relaxation metods, Quart. Appl. Mat., 4 (1946),