Comparative Analysis of the Modified SOR and BGC Methods Applied to the Poleness Conservative Finite Difference Scheme

Transcription

1 EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 6, No. 1, 2013, ISSN Coparative Analysis of the Modified SOR and BGC Methods Applied to the Poleness Conservative Finite Difference Schee Arzu Erde Departent of Matheatics, Kocaeli University, Uuttepe Capus, Izit-Kocaeli, 41380, Turkey Abstract. The poleness conservative finite difference schee based on the weak solution of Poisson equation in polar coordinates is studied. Due to the singularity at r = 0 in the considered polar doain Ω rϕ, a special technique of deriving the finite difference schee in the neighbourhood of the pole point r= 0 is described. The constructed schee has the order of approxiation O (h 2r h2ϕ )/r. In the second part of the paper the structure of the corresponding non-syetric sparse block atrix is analyzed. A special algorith based on SOR-ethod is presented for the nuerical solution of the corresponding syste of linear algebraic equations. The theoretical result are illustrated by nuerical exaples for continuous as well as discontinuous source function Matheatics Subject Classifications: 65M06, 65F50, 65F10 Key Words and Phrases: Finite difference ethod, elliptic proble, polar coordinates, sparse atrix 1. Introduction In this paper we consider the following Dirichlet proble for the Poisson equation in polar coordinates(r,ϕ): Au := r r u 1 2 u = F(r,ϕ), r 2 ϕ 2 (r,ϕ) Ω R, u(r,ϕ)=0, (r,ϕ) Γ ϕ,. (1) u(r,0)=u(r,β), r (0,R), whereω R :={(r,ϕ) R 2 : r [0,R), ϕ [0,β)}, Γ ϕ :={(R,ϕ) : ϕ [0,β)}, and β (0,2π]. This proble is a atheatical odel of various physical and engineering probles arising in steady state flow of an incopressible viscous fluid in a duct of circular cross-section [15, 13], in the deterination of a potential in electrostatics[3] and in the elasticity theory [8]. The two circustances ay lead to singularities: the geoetrical singularity related to Eail address: ÖѺÖÞÙÑкÓÑ 30 c 2013 EJPAM All rights reserved.

2 A. Erde/Eur. J. Pure Appl. Math, 6 (2013), the cornerβ (0,2π) of the polar doain and the pole point r= 0. The first type of singularity eans that for soe values of the paraeter β (0, π) the second (and higher) derivative of the solution u(r, ϕ) with respect to r [0, R) is singular at r = 0. As show exaples below the regularity of the weak solution u H 2 (Ω) H 1 (Ω) of proble (1) depends on the value of the angleβ (0,2π). This behaviour is usual for second order elliptic probles and is related to the C 2 -regularity property of the boundary Ω Rβ of the considered doain[1, 11]. The singularity at the re-entrant corner akes the nuerical solution of these probles challenging. The second type of singularity is a reason of any difficulties in constructing the standard finite difference (FD) schees, when the pole r= 0 is treated as a coputational boundary. To avoid these difficulties various FD and pseudo-spectral (PS) ethods have been suggested in literature[see 4, 14]. These schees include the necessity of special boundary closures, which leads to undesirable clustering of grid points in PS schees[see 4, 6]. The treatent of the singularities related to the situations r 0 and sinϕ 0have been given in[9, 10]. This paper is devoted to fill in the lack of result for conservative finite difference schee for proble (1) and nonsyetric sparse block atrices related to finite difference equations in polar coordinates. We present a conservative finite difference schee for this proble and prove its convergence. Our approach is based on the Lax-Wondroff theore[7], which guarantees convergence of a conservative FD schees in the class of weak solutions, as the polar esh is refined. Note that a siilar technique was used in[12] for proble (1), where the classical solution of the boundary value proble (1) is considered. Since we are interested in bounded weak solutions u H 1 (Ω R ), we require that the solution of proble (1) satisfies the boundedness at r= 0 condition li r u = 0. (2) r 0 Hence one needs to approxiate not only the elliptic equation (1), but also condition (2). This condition will be used for obtaining the conservative finite difference schee in the neighbourhood of the pole point r= 0. Further, the FD approxiations of proble (1)-(2) lead to large sparse syste of linear equations, with the special nonsyetric atrix, due to the periodicity condition u(r, 0) = u(r, β). These type of linear systes require tie-consuing algoriths for their effective nuerical solution[2, 5]. We use the special structure of the obtained atrix[5] and construct a fast iteration algorith, based on SOR-ethod. The paper is organized as follows. In section 2 the weak solution of proble (1)-(2) is defined. The piecewise unifor polar esh and the conservative FD schee for the proble is constructed in Section 3. In Section 4 the structure of the corresponding sparse atrix and iteration algorith is discussed. Nuerical solution of proble (1)-(2) for different types of source function F(r,ϕ) and results are presented in Section 5.

3 A. Erde/Eur. J. Pure Appl. Math, 6 (2013), The Weak Solution and its Regularity Depending on the Paraeterβ (0, 2π) Let v H 1 (Ω Rβ ) be an arbitrary function, where H 1 (Ω Rβ ) :={v H 1 (Ω Rβ ) : u(r,ϕ)=0,ϕ (0,β); u(r,0)=u(r,β), r (0,R)} and H 1 (Ω Rβ ) is the Sobolev space of functions v=v(r,ϕ) with the nor u 1 := Ω Rβ 1/2 u 2 u 2 rdrdϕ. Let us ultiply the both sides of equation (1) to rv(r,ϕ), r (0,β) and integrate onω Rβ : Ω Rβ r u 1 r 2 u ϕ 2 vdrdϕ= F(r,ϕ)v rdrdϕ. Ω Rβ Applying here by part integration, using the boundary and periodicity conditions (2) we get Ω Rβ u(r,ϕ) v(r,ϕ)rdrdϕ= Ω Rβ F(r,ϕ)v rdrdϕ, v H 1 (Ω Rβ ). (3) Here u is the gradient vector in polar coordinates, u= u e 1 1 u r ϕ e e1 Cosϕ, 2, = Sinϕ, e 2 Sinϕ Cosϕ i1 i 2, and i 1, i 2 are unit coordinate vector in Cartesian coordinates. The solution u H 1 (Ω Rβ ) of the integral identity (3) is defined as a weak solution of the boundary value proble (1)-(2). The integral identity (3) shows that if the function u C 2 (Ω Rβ ) C 1 (Ω Rβ ) is the solution of the boundary value proble (1)-(2), then for all v H 1 (Ω Rβ ) this identity holds. Hence the weak solution of proble (1)-(2) can be defined as function u H 1 (Ω Rβ ) satisfying this integral identity for all v H 1 (Ω Rβ ). According to the general theory for linear elliptic boundary value probles the regular weak solution of proble (1)-(2) belongs to H 2 (Ω Rβ ) H 1 (Ω Rβ ), if the boundary Ω Rβ is of class C 2 [4, 11]. The following exaple shows that depending on the valuesβ (π,2π) of the paraeterβ the weak solution of the boundary value proble (1)-(2) ay not belong to the class H 2 (Ω Rβ ). Exaple 1. The function u(r,ϕ)= r π/β Sin(πϕ/β), (r,ϕ) Ω Rβ (4) satisfies the Laplace equation in polar coordinates and the Dirichlet condition u(r,ϕ)=r π/β Sin(πϕ/β), ϕ (0,β).

4 A. Erde/Eur. J. Pure Appl. Math, 6 (2013), This function belongs to the Sobolev space H 2 (Ω Rβ ) H 1 (Ω Rβ ), for allβ (0,π). However for the valuesβ (π,2π) this solution doesn t belong to H 2 (Ω Rβ ), since r α / H 2 (Ω Rβ ) forα<1. The reason, as shown in[11], is that forβ (π,2π) the boundary Ω Rβ doesn t belong to the class C 2, as requires the Agon-Nirenberg regularity theore[1]. Note that the above loss of regularity of the boundary Ω Rβ is a result of the introduced angleα=πϕ/β. When the paraeterβ changes in(0,2π) the angleαalways reains in (0,π). This also iplies that for the function u(r,ϕ), given by (4), the periodicity condition u(r,0) ϕ u(r,β) =, r (0,R) (5) ϕ doesn t hold. The next exaple shows that by introducing the paraeterα= nπϕ/β, n=2, the fulfilent of this condition can be achieved. Exaple 2. Consider now the function u(r,ϕ)= r 2π/β Sin(2πϕ/β), (r,ϕ) Ω Rβ, (6) which evidently belongs to H 2 (Ω Rβ ), β (0,2π). This function satisfies the Laplace equation and the boundary conditions (1). Observe that the angle α = 2πϕ/β always reains in(0, 2π), when the paraeterβ changes in(0,2π). This oent reoves the lack of soothness of the boundary and as a result the solution u(r,ϕ) H 2 (Ω Rβ ) H 1 (Ω Rβ ), β (0,π), given by (6), also satisfies the periodicity condition (5). These two solutions show the ain distinguished features of the boundary value proble (1)-(2), and they will be used for testing of the presented finite difference schee. 3. The Conservative FD Schee on a Piecewise Unifor Polar Mesh We assue hereβ = 2π and introduce the following unifor eshes with respect to variables r and ϕ w r :={r n =(n 0.5)h r : n=1,2,...,n 1, h r =(2R)/(2N 1)}, w ϕ :={ϕ =( 1)h ϕ : =1,2,..., M 1, h ϕ = 2π/M}, with esh steps h r,h ϕ > 0. Then we obtain the piecewise unifor polar esh w rϕ := w r w ϕ (Fig. 1): w rϕ :={(r n,ϕ ) Ω Rϕ : r n w r,ϕ w ϕ }, di w rϕ =(N 1) (M 1), where w rϕ := w rϕ cγ ϕ γ 0, and w rϕ :={(r n,ϕ ) Ω Rϕ : n=2, N, =2, M}. The boundary esh points are defined as follows γ ϕ :={(R,ϕ ) Γ ϕ : =1, M},γ 0 :={(r n,0) Γ 0 : n=1, N}.

5 A. Erde/Eur. J. Pure Appl. Math, 6 (2013), rotated esh γ φ γ β Figure 1: Geoetry of the poleness polar esh and its rotated for Due to the periodicity condition u(r,0)=u(r,2π) we will not include the values u(r,0), r [0,R] to the list of unknowns in the discrete proble. The introduced esh w rϕ doesn t include the pole point r= 0, that is in the presented discrete odel the doain Ω Rϕ is approxiated by the circular discω o Rϕ. Thus our discrete odel does not deal with the singularity at r= 0, that is usual for the differential proble. Instead we will derive an approxiation of the boundedness condition (2) at the central circle with radius r=r 1, r 1 = 0.5h r. Denote by e n ={(r,ϕ) Ω Rβ : r n r r n1,ϕ ϕ ϕ 1 } the polar finite eleent with four nodes. We derive an error approxiation for each eleent. Introducing the halfnodes r ± n = r n± h r /2,ϕ ± =ϕ ± h ϕ /2 and integrating equation (1) on the finite eleent e n :=[r n, r n ] [ϕ,ϕ ] we obtain the following balance equation: ϕ r n ϕ rn r u r n ϕ 1 2 u drdϕ r rn ϕ ϕ 2 dϕdr= r n r n ϕ ϕ rf(r,ϕ)dϕdr. (7) Let us transfor the first left integral I r n. I r n = ϕ ϕ r u r=r n r=rn dϕ = h ϕ r u (r n,ϕ ) r u (r n,ϕ ) We use here the central finite difference forula for approxiation of derivatives on the right hand side, by using the esh points r n, r n1/2, r n1, with esh step h r/2 = h r /2: r u r u (r n,ϕ u(r ) = r n1,ϕ ) u(r n,ϕ ) n, h r (r n,ϕ ) = r n u(r n,ϕ ) u(r n 1,ϕ ) h r.

6 A. Erde/Eur. J. Pure Appl. Math, 6 (2013), Then we have the following variational finite difference approxiation of the integral operator I r n : I r n= h ϕ r n u r,n r n u r,n. By the sae way we can derive an approxiation of the second integral operator I ϕ n on the left hand side of (7): I ϕ n u = h r u r n ϕ (r n,ϕ ) u ϕ (r n,ϕ ) = h r r n uϕ (r n,ϕ ) u ϕ (r n,ϕ ). Applying to the right hand side of (7) the nuerical integration (rectangle) forula, finally we have h ϕ [r n y r,n r n y r,n] h r r n [y ϕ,n y ϕ,n ]=h r h ϕ r n F(r n ϕ ) Dividing by h r h ϕ r n > 0 we obtain the following finite difference equation 1 1 n u := r (r y r) r,n r 2 y ϕϕ = F(r n,ϕ ), (r n,ϕ ) ω rϕ, n 1. (8) n The finite diensional operators r n y := 1 r (r y r) 1, ϕ n y r,n r 2 y ϕϕ are the finite difference approxiations of the differential operators A r u := 1 r n k(r) u, A ϕ u := 1 2 u r 2 ϕ 2,(r,ϕ) Ω Rβ, correspondingly. Note that the sae approxiations can also be obtained fro the direct finite difference approxiation of the Poisson equation (1). The finite difference equation corresponding to the layer r=r 1 = h r /2 can be derived by using the sae balance equation (7), substituting r n =ǫ, r n = h r: ϕ hr ϕ ǫ r u ϕ hr 1 2 u drdϕ r ϕ ǫ ϕ 2 drdϕ= ϕ hr ϕ ǫ rf(r,ϕ)drdϕ. Going to the liitǫ 0 here and using condition (2) we obtain ϕ ϕ u(h r,ϕ) dϕ hr 0 1 r u(r,ϕ ) u(r,ϕ ) hr ϕ dr r F(r,ϕ)dϕdr= 0. ϕ ϕ 0 ϕ

7 A. Erde/Eur. J. Pure Appl. Math, 6 (2013), Again using the nuerical integration forula in the first integral we get u(h r,ϕ ) u hr h r h ϕ 2 ϕ 2,ϕ h ϕ u hr 2 ϕ 2,ϕ h ϕ h2 r h ϕ hr 2 2 F 2,ϕ 0 To derive this finite difference equation in canonical for we use the standard definitions Then we have y r,1 := 1 h r [y(r 1 h r,ϕ ) y(r 1,ϕ )], y ϕ,1 := 1 h ϕ [y(r 1,ϕ ) y(r 1,ϕ 1 )], y ϕ,1 := 1 h ϕ [y(r 1,ϕ 1 ) y(r 1,ϕ )]. h r h ϕ y r (r 1,ϕ )2h ϕ y ϕ,ϕ (r 1,ϕ ) h2 r h ϕ 2 F(r 1,ϕ )=0, =1,2,..., M Dividing the both sides to h 2 r h ϕ/2 finally we obtain the finite difference equation corresponding to the layer r 1 = h r /2: 2 y r (r 1,ϕ ) 4 h r h 2 y ϕ,ϕ (r 1,ϕ )= F(r 1,ϕ )=0, =1,2,..., M. (9) r Equations (8)-(9) represent the finite difference analogue of the Poisson equation (1) in the constructed polar eshω rϕ. Lea 1. If u C 4 (Ω Rβ ) then the order of the approxiation error of the finite difference schees (8)-(9) in C-nor isψ rϕ :ψ rϕ = O (h 2 r h2 ϕ )/r n, where ψ rϕ = h2 r 6r 3 u n h r /2 3r 4 4 u(r n,ϕ ) 4 r n h r /2 4 4 u(r n,ϕ) h2 ϕ 4 u(r n,ϕ ) 4 12r ϕ 4. The explicit for (10) ofψ(h r,ϕ ) and the proof of this result is given in[11]. The lea shows that as r r 1 the functionψ rϕ increases, and at the first layer r=r 1 (r 1 = h r /2) becoesψ rϕ = O(h r h 2 ϕ /h r). Note that the boundedness condition (2) is necessary for the above approxiation, and hence for the convergence of the finite difference schees (8)-(9), although this condition is not used explicitly on deriving the approxiation error. To show this, consider the following: Exaple 3. The function (10) u(r,ϕ)=ln 1 r Sinϕ, (r,ϕ) Ω R,ϕ,

8 A. Erde/Eur. J. Pure Appl. Math, 6 (2013), satisfies the Poisson equation (1) inω Rβ, with for R=1,β = 2π, and the right hand side F(r,ϕ)= 1 r 2 Sinϕ. Evidently this solution also satisfies the boundary and periodicity conditions (1), but does not satisfy the boundedness condition (2), since r u/= Sinϕ 0, as r 0. Calculating the right hand side of (10) for n=1 we obviously observe that for r=r 1 ψ(r 1,ϕ )= h2 ϕ 12r 3 sinϕ. 1 This shows thatψ(r 1,ϕ ) 0, as r 1 0, and there is no approxiation in the neighbourhood of the pole point r= 0. To forulate the discrete proble we need to add to equations (8)-(9), the equations, obtained fro the periodicity condition (1). 4. The Algebraic Proble with Nonsyetric Sparse Matrix: Iteration Algorith The finite difference equations (8)-(9) copose K := N M nuber of algebraic equations with K unknowns y=(y 11, y 12,..., y 1M, y 21,..., y N M ) T, di y= K. We can rewrite these equations in the for of the syste of linear algebraic equations y= with the following positive band atrix, di= K K: := A 11 A A 21 A 22 A A 32 A 33 A A N 1,N 2 A N 1,N 1 A N 1,N A N,N 1 A N,N Here M M-diensional block atrices A i j are of the following structure: A ii = and A ii1 =α i I, A i 1i =β i I.

9 A. Erde/Eur. J. Pure Appl. Math, 6 (2013), The paraetersα i andβ i can be defined fro the finite difference schees (8)-(9) as follows: α 1 = 2 h 2 ; α i = k i r r i h 2 r, i= 2, N; β i = k i1 r i h 2, i= 1, N. r Sinceα i β i, the atrix is not syetric one. Moreover, is a sparse atrix of special structure, corresponding to the polar esh and periodicity condition. Evidently such a syste of linear algebraic equations with non-syetric sparse atrix needs to be solved by iteration ethods[12, 13, 14]. However, as the coputational experients below show, use of copact storage for the band atrix and then an application of any effective iteration ethod requires large enough tie for the solution of the linear syste of algebraic equations y=. The reason is that the bandwidth of the non-syetric atrix is bw= 3M and there are any null ters in the band. Specifically, the above block atrices A ii, A ii1, A i 1i contain M 2 3M, M 2 M, M 2 M zero eleents, correspondingly. Hence for M= N the nuber of zero and non-zero eleents of the band are 3N 3 4N 2 4N and 5N 2 N, respectively. For the esh with N = 30, this eans that the nuber of non-zero eleents of the band atrix is about 4.3% of all band eleents. Therefore one needs to construct a special algorith which can store and operate with only nonzero eleentsα i andβ i, realising the ultiplications U y and Ly, to iniize the tie required for the solution of the considered proble by any iteration ethod. Here the U and L are the upper and lower triangular atrices:= DUL. Note that for soe class of systes, arising fro the finite-eleent discretization, siilar algorith was constructed in[15]. Table 1 illustrates the coparative analysis of the standard SOR ethod y k1 =(DωL) 1 [(1 ω)d ωu]y k ω(dωl) 1, (11) by using MATLAB code, and SOR ethod with the constructed here special algorith. As a test exaple the analytical solution given in Exaple 2, withβ= 2π, is used. The iteration paraeterω (0,2) in (11) was defined by the forula 2 ω= 1, λ in (2 λ in ) whereλ in is the inial eigenvalue of the Laplace operator, andλ in = 2sin 2 (π/2n), for the square esh M= N. Table 1 shows that direct application of the SOR ethod is expensive in the sense of the required CPU tie. This tie increases as the nuber K= N M of esh points increases. Coputational results show that this increase has the character 2 n, i.e. n-ties increase of the nuber of esh points leads 2 n -ties increase of the CPU tie. This is due to the oderately ill-conditionedness, according to[12], of the atrix, as the fourth colun of the table shows. At the sae tie, as the second colun of Table 1 shows, the SOR ethod with the special algorith requires less than 1 second for all considered eshes.

10 A. Erde/Eur. J. Pure Appl. Math, 6 (2013), Table 1: Coparison of the standard and odified SOR algoriths N M SOR with special algorith Standard SOR Condition nuber CPU tie(sec.) CPU tie(sec.) of the atrix Nuerical Solution of the Elliptic Proble (1)-(2) by the Poleness Conservative Schees (8)-(9) In this section we discuss results of coputational experients related to nuerical solution of the Dirichlet proble (1)-(2) by the poleness conservative schee (8)-(9). In the first series of the coputational experients, the convergence and accuracy of the nuerical solution of the Dirichlet proble (1)-(2), with the analytical solution u(r,ϕ)=(r 1)Sinϕ, (r,ϕ) Ω Rβ, R=1, is studied. Note that the corresponding source function F(r,ϕ)= 1 r 2 Sinϕ, (r,ϕ) Ω Rβ, (12) has singularity at r = 0. The two appropriate iterative ethods - SOR ethod and Bi- Congugate Gradient (BCG) ethod - are applied for the iterative solution of the linear syste of algebraic equations, corresponding to the finite difference schees (8)-(9). In all cases the MATLAB codes of this ethods with the above entioned special algorith is used. Results are presented in the Table 2. For the coparison, the nuerical results obtained by the Gauss- Seidel ethod, are also presented. Here and below the value of the stopping paraeterδ>0 in u ni 1 u n i 0 δ, where 0 is the L 2 -nor, is takenδ=10 5. Table 2 shows nubers of iterations corresponding to all three iteration ethods, with H 1 -relative and L -absolute errors. Results given in the table show that for relatively coarse eshes BGC ethod is ore effective than the SOR ethod. But for the eshes and higher, SOR ethod is ore effective in the sense of iterations. Moreover, the nuber of iterations n i in SOR ethod increases slowly, as increases the nuber of esh points. Thus n i = 234 and n i = 277, for the eshes and 50 50, respectively. As show the last three coluns of the table, the H 1 -relative and L -absolute errors are sall enough, although the source function (12) has singularity at the pole point r= 0. The next series of coputational experients is realized for the sooth continuous source function 50ex p( ǫ2 ), 0< r<ǫ F(x, y)= F(r,ϕ)= ǫ 2 r 2 (13) 0, ǫ<r< R, R=1,

11 A. Erde/Eur. J. Pure Appl. Math, 6 (2013), Table 2: Coparative analysis of the odified SOR and BGC ethods applied to the poleness conservative finite difference schee (8)-(9) Methods N M Tie Nuber of Rel. error Abs. error Abs. error (sec.) iterations H 1 -nor L -nor L -nor n i r=r 1 r= R/ SOR BCG Gauss Seidel withǫ= 1/2, approxiating in weak sense the Diracδ-function (Figure 2a). This function is taken as a given data for the Dirichlet proble (1)-(2). The right pane, Figure 2b, illustrates the nuerical solution ũ h (x, y)=u h (r,ϕ) of proble (1)-(2) by the poleness conservative schees (8)-(9), for the esh size Finally we consider the weak solution of the Dirichlet proble (1)-(2), when the source F(r, ϕ) is a discontinuous at r = 1/2 function F(x, y)= F(r,ϕ)= 50ǫ exp( ǫ2 /4r) 4πr, 0< r<ǫ 3 50ǫ exp( ǫ2 /4) 4π, ǫ<r< R, R=1, given in the left pane of Figure 3. This function withǫ = 1/2 is taken as a given data for the Dirichlet proble (1)-(2). The nuerical solution ũ h (x, y)=u h (r,ϕ) of proble (1)-(2) obtained for the esh is plotted in the right pane, Figure 3b. To estiate an accuracy of the nuerical solution, in particular at the discontinuity point r = 1/2, the nuerical solutions u (1) h (r,ϕ)=u(2) (r,ϕ) corresponding to two different eshes w(1) h rϕ = w(2) rϕ is copared. The relative error u (1) ǫ h = h (r,ϕ) u(2) h (r,ϕ), 0.5 u (1) h (r,ϕ)u(2) h (r,ϕ) is aboutǫ h = , including the discontinuity point. This shows high accuracy of the nuerical ethod in the case of discontinuous source function, also.

12 A. Erde/Eur. J. Pure Appl. Math, 6 (2013), (a) Continuous Source Solution (b) Nuerical Solution Figure 2: Dirichlet proble in polar coordinates (a) Discontinuous Source Solution (b) Nuerical Solution Figure 3: Dirichlet proble in polar coordinates

13 REFERENCES Conclusion We studied poleness conservative finite difference schee for Laplace operator in polar coordinates. The schee with the odification of the SOR ethod allows to construct an effective nuerical ethod for solving the Dirichlet proble in the polar coordinates, based on the weak solution approach. Nuerical results presented for discontinuous source function shows high accuracy of the ethod on acceptable eshes. Extension of results given here can be ade for positive elliptic operators with discontinuous coefficients, and for nonlinear onotone operators of Plateau type, as well. This require soe additional techniques that will be done in next studies. References [1] S. Agon, A. Douglis, and L. Nirenberg. Estiates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. Counications on Pure and Applied Matheatics. 17, [2] A.M. Bruaset. A Survey of Preconditioned Iterative Methods. Addison-Wesley [3] J.D. Jackson. Classical Electrodynaics. 2nd Ed., Wiley, New York [4] M.D. Griffin, E. Jones, and J.D. Anderson. A coputational fluid dynaic technique valid at the centerline for non-axisyetric probles in cylindrical coordinates. Journal of Coputational Physics. 30, [5] W. Hackbush. Iterative Solution of Large Sparse Systes of Equations. Springer-Verlag, Berlin [6] W. Huang and D.M. Sloan. Pole condition for singular probles: The pseudo-spectral approxiation. Journal of Coputational Physics. 107, [7] P.D. Lax and B. Wendroff. Systes of conservation laws. Counications on Pure and Applied Matheatics. 13, [8] A.I. Lurie. Three-Diensional Probles of the Theory of Elasticity. Interscience Publishers, New York [9] P.E. Merilees. The pseudospectral approxiation applied to the shallow water equations on a sphere. Atosphere, 13(1), [10] K. Mohseni and T. Colonius. Nuerical treatent of polar coordinate singularities. Journal of Coputational Physics. 157, [11] M. Renardy and J. Rogers. Introduction to Partial Differential Equations. Springer-Verlag, New York, 1993.

14 REFERENCES 43 [12] A.A. Saarskii and V.B. Andreev. Difference Methods for Elliptic Probles (in Russian). Nauka, Moscow [13] F.S. Sharan. Viscous Flow. McGraw-Hill, New York [14] R. Verzicco and P. Orlandi. A finite difference schee for three diensional incopressible flows in cylindrical coordinates. Journal of Coputational Physics. 123, [15] F.M. White. Viscous Fluid Flow. McGraw-Hill, New York