An Iterative Method for Finite-Element Solutions of the Nonlinear Poisson-Boltzmann Equation

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1 WEA TRANACTION on COMPUTER An Iteratve Method for Fnte-Element olutons of the Nonlnear Posson-Boltzmann Equaton Natonal Kaohsung Normal Unversty Department of Mathematcs No. 6, Ho-png st Rd., Lngya Dstrct, Kaohsung Cty 8 Tawan (R.O.C.) rcchen@nnucc.nnu.edu.tw Abstract: A fnte-element (FE) approach combned wth an effcent teratve method have been used to provde a numercal soluton of the nonlnear Posson-Boltzmann equaton. The teratve method solves the nonlnear equatons arsng from the FE dscretzaton procedure by a node-by-node calculaton. Moreover, some extensons called by Pcard, Gauss-edel, and successve overrelaxaton (OR) methods are also presented and analyzed for the FE soluton. The performances of the proposed methods are llustrated by applyng them to the problem of two dentcal collodal partcles n a symmetrc electrolyte. My numercal results are found n good agreement wth the prevous publshed results. A comprehensve survey s also gven for the accuracy and effcency of these methods. Key Words: fnte-element method, Posson-Boltzmann equaton, collodal partcles nteracton Introducton The nonlnear Posson-Boltzmann (PB) equaton descrbes, n some approxmaton, the electrc potental and charge dstrbuton n collodal systems, ]. Knowng the electrostatc potental, one can calculate other quanttes such as the free energy of a collodal system and the force of partcle-partcle nteracton. Features of nter-partcle nteracton are of great mportance n studyng the stablty of collodal dspersons, the formaton of collodal crystals and membrane separaton processes 3]. To obtan numercal solutons of the PB equaton, one must solve a system of nonlnear algebrac equatons resultng from a dscretzaton by, for example, the fnte-element (FE) method. The standard method for the soluton s Newton s method or ts varant 4]. Newton s method s a local method that converges quadratcally n a suffcently small neghborhood of the exact soluton. It s very senstve to ntal guesses due to ts local convergence property. We propose here an teratve method whch s globally and monotoncally convergent wth smple upper or lower solutons of the PB equaton as ntal guesses 5]. The method of monotone teratons s a classcal tool for the study of the exstence of the soluton of semlnear partal dfferental equatons of certan types 6, 7, 8, 9]. It s also useful for the numercal soluton of these types of problems approxmated, for nstance, by the fnte dfference,, ], fnte element 3], or boundary element 4, 5, 6] method. IN: It s a constructve method that depends essentally on only one parameter, called the monotone parameter heren, whch determnes the convergence behavor of the teratve process. Embedded n the wdely used FE algorthm, four monotone teratve methods, namely, Jacob, Pcard, Gauss-edel, and OR methods are presented and analyzed n ths paper. In the next secton, we state the model problem from the collodal system to the correspondng PB equaton and FE algorthm. The model s subject to Drchlet and Neumann types of condtons on varous parts of the boundary of an rregular doman. tartng wth the upper soluton as an ntal guess, t s shown n ecton 3 that maxmal sequences generated by Jacob, Pcard, Gauss-edel, and OR teratons all converge monotoncally from above to the unque soluton of the resultng nonlnear system. In ecton 4, we represents a part of our extensve numercal experments on varous confned and unconfned models to demonstrate the accuracy and effcency of monotone propertes of the proposed methods. Moreover, a short concludng remar s gven n ecton 5. Descrpton of the Problem Owng to the symmetry, all of the problems consdered have the same two-dmensonal doman Ω whch s shown n Fg.. egment CD s the wall of a cylndrcal vessel, segment DE s the outlet, segment BC represents a mdplane for the problems wth two par- Issue 4, Volume 7, Aprl 8

2 WEA TRANACTION on COMPUTER C R D The dmensonless force obtaned by ntegratng over the surface of the partcle s calculated accordng to the expresson B A κ a G Fgure : The doman for the problem of two nteractng dentcal sphercal partcles. tcles, segment AB s half the separaton dstance L and segment BE s the axs of rotatonal symmetry. The dmensonless PB equaton for electrostatc potental Ψ outsde the spheres n cylndrcal coordnates taes the form Ψ R + Ψ R R + Ψ = snh Ψ () Z Length, electrostatc potental, and force are respectvely measured n unts of Debye length κ = (nqe/ɛt ) /, T/q e, and ɛ(t/q e ), where n s the concentraton of any of the speces n the electrolyte, q e s the absolute value of electronc charge, ɛ s the absolute permttvty of the electrolyte, s the Boltzmann constant, T s the absolute temperature, and the ratonalzed I s used to express the factors. To solve Eq. (), t s necessary to defne the potental Ψ and the potental gradent Ψ/ N at the boundary Ω of the doman Ω. N (=κn) s the outward normal drecton from the boundary Ω. In the present paper, Ψ = Ψ s on the boundary AGF (whch refers to the sphere surface), and Ψ = Ψ m on the boundary CD (whch refers to the pore wall). Axs symmetry of the geometry mples that dervatves wth respect to the coordnates R on the lne EF are assumed to be zero. Also, on the lne DE natural boundary condton of the Neumann type s satsfed. The electrc potentals wthn the spheres and the materal surroundng the pore are constant. The electrc feld s related to the potental by the equaton E = Ψ. The force of nteracton of the partcles s obtaned by means of drect ntegraton of the total stress tensor over the approprate surface. There are at least two possble ways of ntegratng: over the surface of the partcle and over the mdplane. IN: 9-75 F Z E 66 π F s = (κ a ) Ψ cos θ sn θ dθ, () where κ a s the dmensonless sphere radus. For the ntegraton over the mdplane, say M, the dmensonless force s ( ) Ψ F m = (cosh Ψ ) + M R ( ) ] Ψ R dr, (3) Z The latter case s more accurate snce dfferent peces of the mdplane contrbute wth the same sgn. 3 Monotone Iteratve Methods Let T be a FE partton of the doman Ω such that T = { τ j : j =,..., M, Ω = M j= τ j} and the system of nonlnear algebrac equatons resultng from FE dscretzaton s η ψ η ψ = R (ψ ) + R (4) V () where the set V () of degrees of freedom satsfes η, V (),, the functon R(, Ψ) s nonlnear n Ψ descrbng the PB equaton and R s prescrbed n the boundary Ω. The dagonal domnance of the resultng matrces (.e., M-matrces 7]) of the model problems provdes not only stablty of numercal solutons (.e., no non-physcal oscllatons) but also convergence of teratve procedures. Ths s a basc hypothess for the development of varous monotone teratve schemes for (4). Defnton A vector Ψ ( ψ,..., ψ N ) IR N s called an upper soluton of (4) f t satsfes the followng nequalty η ψ V () η ψ R ( ψ ) + R, (5) and ˆΨ ( ˆψ,..., ˆψ N ) IR N s called a lower soluton f η ˆψ V () η ˆψ R ( ˆψ ) + R, (6) for N where N s the total number of node ponts. Issue 4, Volume 7, Aprl 8

3 WEA TRANACTION on COMPUTER Gven any ordered lower and upper solutons ˆΨ and Ψ, we defne the soluton sectons by ˆΨ, Ψ {Ψ IR N ; ˆΨ Ψ Ψ} ˆψ, ψ {ψ IR; ˆψ ψ ψ }. 3. Jacob Method Now we ntroduce the maxmal sequence. Let V () = Ψ be an ntal terate. We construct a sequence {V (m+) } by solvng the lnear system η v (m+) V () η v (m) + γ (m+) v (m+) = γ (m+) v (m) R ( v (m) ) + R, (7) for m =,,,..., N and the monotone parameter γ (m+) s defned by γ (m+) R ( v (m) ) ψ. (8) For the maxmal sequence we have the followng propertes 8, ]. Lemma Assume the nonlnear functon R (ψ ) s monotone ncreasng and concave up wth respect to ψ,.e., R / ψ >. Then the maxmal sequence { } gven by (7) wth V () = Ψ possesses the monotone property ˆΨ V (m+) Ψ, m =,,,.... (9) Moreover, for each m, s also an upper soluton. Proof. Let e () v () v () = ψ v (). By (7) we have (η + γ () )e () = (η + γ () ) ψ η v () + γ () v () R (v () ) + R ] V () = η ψ In vew of e () V () η ψ + R ( ψ ) R. for all N. Ths leads to V () Ψ. Assume, by nducton, that v (m) some m >. By (7) e (m) v (m) v (m+) v (m) for satsfes (η + γ (m+) )e (m) = η (v (m) v (m) ) V () R (v (m) ) + R (v (m) ) + γ (m) γ (m) v (m). IN: 9-75 v (m) 67 By R/ Ψ and the mean value theorem, we have R (v (m) ) R (v (m) ) R (v (m) ) ψ (v (m) v (m) ), R (v (m) ) R (v (m) ) γ (m) v (m) γ (m) v (m). Ths yelds e (m) whch shows that v (m+) v (m) and hence monotone property (9) thus follows by nducton. To show that ˆΨ V (m+), we assume, by nducton, that ˆψ v (m) for some m >. By (6) and (7) we have (η + γ (m+) + + V () V () γ (m+) )(v (m+) η v (m) η (v (m) ˆψ ) v (m) ˆψ ) = (η + γ (m+) ) ˆψ + γ (m+) v (m) R (v (m) ) + R ] γ (m+) ˆψ + R ( ˆψ ) R (v (m) ] ). By the concave up property of R (u ) and the mean value theorem agan, we have γ (m+) v (m) γ (m+) ˆψ + R ( ˆψ ) R (v (m) ). Ths yelds (η + γ (m+) shows )(v (m+) v (m+) ˆψ. ˆψ ) whch To show that V (m+) s an upper soluton for each m, we observe from (7) and the monotone property of { } that η v (m+) = V () + γ (m+) V () η v (m) v (m) γ (m+) v (m+) R ( v (m) ) + R, η v (m+) + γ (m+) + γ (m+) v (m) R ( v (m) ] ) + R. v (m+) By the concave up property of R (u ) and the mean value theorem, we have γ (m+) v (m) v (m+) ] R ( v (m) ) R ( v (m+) ). Therefore, η v (m+) V () η v (m+) R ( v (m+) ) + R. Ths shows that s an upper soluton. Issue 4, Volume 7, Aprl 8

4 WEA TRANACTION on COMPUTER Theorem 3 Assume condtons n Lemma hold. Then the sequence { } generated by solvng (7) wth V () = Ψ converge monotoncally to the soluton V of (4). Moreover ˆΨ V V (m+) Ψ, m =,,.... () and f Ψ s any soluton of (4) n ˆΨ, Ψ then ˆΨ Ψ V. Proof. By Lemma and the completeness property, the lmt lm m = V exsts and satsfes the relaton (). Now f Ψ ˆΨ, Ψ s a soluton of (4) then Ψ and Ψ are ordered upper and lower solutons. Usng V () = Ψ, Lemma mples that Ψ for every m. Lettng m gves V Ψ. Ths proves the theorem. 3. Pcard and Gauss-edel Methods Let A be the matrx obtaned by FE dscretzaton. It can be wrtten n the splt form A = D L U, where D, L and U are the dagonal, lower-off dagonal and upper-off dagonal matrces of A, respectvely. The elements of D are postve and those of L and U are nonnegatve. Usng ˆΨ and Ψ as the ntal terates we can construct the three maxmal sequences by the three teratve schemes defned as follows: (a) Pcard method (A+Γ (m+) P )V (m+) P = Γ (m+) P P R( P ) + R, () (b) Gauss-edel method (c) Jacob method where Γ (m+) P Γ (m+) G (D L+Γ (m+) G )V (m+) G = U G +Γ (m+) G (m) G R(V G ) + R, () (D+ Γ (m+) J )V (m+) J = (L + U) J +Γ (m+) J J R( J ) + R, (3) dag(γ (m+) P, ), γ (m+) P, dag(γ (m+) G, ), γ (m+) G, IN: 9-75 R (v (m) P, ) ψ, R (v (m) G, ) ψ, (4) (5) 68 Γ (m+) J dag(γ (m+) J, ), γ (m+) J, R (v (m) J, ) ψ, (6) and the ntal guesses are V () P = V () G = V () J = Ψ. The followng lemma gves the monotone property of these sequences. Lemma 4 Assume the condtons n Lemma hold. Then the maxmal sequence { } gven by ether one of the teratve schemes ()-(3) wth V () = Ψ possesses the monotone property (). P G for every m =,, 3,.... overrelax- 3.3 OR Method Typcally for the lnear system Ax = b aton s base on the splttng J, (7) ωa = (D ωl) ( ω)d + ωu], Theorem 5 Assume the condtons of Lemma hold. Then each of the maxmal sequences G, J, P converges monotoncally to the soluton V of (4) and satsfes the relaton (). Moreover, and the correspondng successve overrelaxaton (OR) method s gven by the recurson (D ωl)x (m+) = ( ω)d + ωu]x (m) + ωb, where ω s called the acceleraton parameter. For the nonlnear system solved by the monotone teratve method we defne (D + Γ (m+) ωl)v (m+) = ( ω)(d + Γ (m+) ) + ωu + ω Γ (m+) where the monotone parameter Γ (m+) Γ (m+) dag(γ (m+), ), γ (m+), ] R( ) + R ],(8) s defned by R (v (m), ) ψ. (9) Lemma 6 Assume the condtons n Lemma hold. Moreover, f < ω. () Then the maxmal sequence { } gven by the teratve scheme (8) wth V () = Ψ possesses the monotone property (). Moreover, for each m, s also an upper soluton. Issue 4, Volume 7, Aprl 8

5 WEA TRANACTION on COMPUTER Proof. Let E () V () V () = Ψ V (). By (8) we have (D + Γ () ωl)e() = (D + Γ () ωl)( Ψ V () ) = (D + Γ () ωl) Ψ = ω ( ω)(d + Γ () ] ) + ωu V () ω Γ () V () () R(V ) + R ] (D L U) Ψ + R( Ψ) R ]. Here Ψ s an upper soluton ω > and (D + Γ () ωl) exsts and s nonnegatve (cf. 7]). Therefore E (),.e.,v () Ψ. Assume, by nducton, that some m >. By (8) E (m) satsfes (D + Γ (m+) ωl)e (m) = (Γ (m+) + (D+Γ (m) (m) ωl)v V (m) V (m+) for Γ (m) (m) )V (D+Γ(m+) ωl)v (m+) (m) )V = (Γ (m+) Γ (m) + ( ω)(d + Γ (m) ] ) + ωu V (m) + ω Γ (m) V (m) R(V (m) ) + R ] ( ω)(d + Γ (m+) ] ) + ωu ω Γ (m+) R( ) + R ] = (Γ (m+) Γ (m) (m) )V + ( ω)d + ωu] (V (m) ) + Γ (m) V (m) + ωr(v (m) )] Γ (m+) ωr( )] = ( ω)d + ωu] (V (m) ) + ω Γ (m) V (m) R(V (m) ) Γ (m) + R( ] ). mlarly, we can use (), concave up property of R( ) and the mean value theorem to have (D + Γ (m+) ωl)e (m). Therefore E (m),.e.,v (m+). To show that V (m+) s an upper soluton, we consder (8), (D + Γ (m+) ωl)v (m+) = (D + Γ (m+) + ω Γ (m+) ) + ω(d Γ (m+) R( ) + R ]. After movng the terms, (D + Γ (m+) rght-hand sde and usng (), we obtan ωlv (m+) = DV (m+) +Γ (m+) +ω Γ (m+) ( D V (m+) ( )V (m+) + ( ω)d V (m+) ) + ω + ( ω)d R( + U), to the + ωu ) + R ] + ωu V (m+) ) R( ) + R ]. By the mean value theorem and have ωlv (m+) DV (m+) + ωuv (m+) V (m+) + ( ω)d + ω we R(V (m+) ) + R ] After addng the term, ωdv (m+) and usng the fact ω, we obtan ω(d L)V (m+) ( ω)d( V (m+) ) +ωuv (m+) + ω R(V (m+) ) + R ] ωuv (m+) + ω R(V (m+) ) + R ]. Therefore, we move the term, ωuv (m+), to the lefthand sde and cancel the ω to have (D L U)V (m+) Ths proves that V (m+) R(V (m+) ) + R. s an upper soluton. 4 Results and Dscussons 4. Interacton of Two Identcal Charged phercal Partcles Ths problem deals wth two dentcal collodal partcles mmersed n symmetrcal : electrolyte. It was studed n several wors and can serve as a test, 9, ]. In the present paper, the force of nteracton of two partcles of the radus κ a =. and 5. were calculated for the separaton dstance L =. and.5 respectvely. The constant potental Ψ on the surfaces of both partcles was equal to.. The Neumann boundary condtons Ψ/ n = are mpled on the other boundares of the doman. A typcal mesh and soluton are shown n Fg. and 3 for a case of two nteractng sphercal partcles wth κ a = 5, Ψ s = and L =.5. Table shows re- IN: Issue 4, Volume 7, Aprl 8

6 WEA TRANACTION on COMPUTER R axs sults for the dmensonless electrostatc force between two dentcal sphercal partcles for gven condtons, whch compared wth some prevously publshed papers. The results are n good agreement of ours. Table The Dmensonless Force between Two Partcles κ a F m F p F p F p Note that F m s the force on the mdplane, F p s the force from prevous results ], F p s the force from prevous results ], and F p3 s the force from prevous results 9]. Z axs Fgure : The mesh for the problem of two nteractng dentcal charged sphercal partcles Z axs R axs Fgure 3: The numercal soluton for the problem of two nteractng dentcal charged sphercal partcles. 4. Interacton of Two Identcal Charged phercal Partcles Confned wthn a Charged Cylndrcal Pore Ths problem deals wth the long-range electrostatc nteracton of two charged spheres confned n a lecharged cylndrcal pore. The same parameters are used, e.g., the : electrolyte, the constant potental on the cylndrcal pore Ψ P = 5., and the constant potental on the spheres Ψ = 3.. The radus of the partcles s κ a =.85 and the sphere radus to pore radus rato s λ =.3. Fg. 4 shows the sopotental plot for two solated spheres (Ψ = 3.) and two spheres confned n a pore (Ψ = 3. and Ψ P = 5.). They are found n good agreement wth the publshed results, see, e.g., ]. Fg. 5 shows the numercal soluton for the confned case. In order to observe the behavor of the error reducton for varous teratve schemes the error e (m) v (m) v (m) s defned and the stoppng crteron for these teratons s determned from the condton e (m).e6. Fg. 6 shows the typcal phenomena of monotone convergence n varous schemes for the confned case (Ψ = 3. and Ψ P = 5.). The behavors of the resdual, A +R( ), of varous methods are shown n Fg. 7. nce the soluton fgure and the convergence behavor are smlar, we sp the unconfned case (Ψ = 3.). Table The Nt and CPU Tme Versus Varous Methods Ψ = 3. Pcard (G) Pcard () G Jacob OR nt Tme (ec) Ψ = 3. Pcard (G) Pcard () G Jacob OR Ψ P = 5. nt Tme (ec) In the table, nt denotes the number of teratons, Tme s the CPU tme (Intel Pentnum D 8), Pcard IN: Issue 4, Volume 7, Aprl 8

7 WEA TRANACTION on COMPUTER R axs Resdual 3 4 Z axs R axs Iteraton No. Fgure 6: The resdual versus the number of teratons for the case of two solated spheres. old lne: Jacob method, dotted lne: Gauss-edel method and dashed lne: Pcard method. old lne wth trangles: OR method wth ω=.8, old lne wth crcles: OR method wth ω=.6. Z axs Fgure 4: Calculated sopotental lnes for the problem consdered n subsecton 4.. A half-secton of the physcal geometry s shown, wth the lne of symmetry lyng at the bottom. On the top of the graph, solated spheres; on the bottom of the graph, spheres confned n a pore. The pore wall s at the top. (G) means Pcard method wth Gaussan elmnaton for the lnear solver, Pcard () means Pcard method wth Gauss-edel method for the lnear solver, G s the Gauss-edel method and OR s successve overrelaxaton method wth ω =.6. The convergence of the Pcard method s the fastest, then the OR method, and then the Gauss-edel method and the Jacob method follows accordngly. On the one hand the teratve behavor of the Pcard method s remarable for ts fast convergence. It s fnshed after seventh (Ψ = 3.) and nnth (Ψ = 3. ΨP = 5.) teratve step and more faster than Gauss-edel and Jacob methods. Ths phenomenon verfes Theorem 5. On the other hand the memory storage and the CPU tme consumng n Gaussan elmnaton are the drawbacs of the Pcard method. Fg. 8 shows the number of teratons versus the acceleraton parameter for the case of spheres confned n a pore. The best value of ω s R axs 5 Z axs Fgure 5: The numercal soluton for the case of spheres confned n a pore. IN: Remar 7 In lterature, reorderng rows and columns s one of mportant ngredents used n parallel mplementatons of both drect and teratve soluton technques. The type of reorderng used n applcatons depends on whether a drect or an teratve method s beng consdered. In 97, George 3, 4] observed that reversng the Cuthll-McKee (RCM) orderng yelds a better scheme used to enhance the effectveness of sparse Gaussan elmnaton. We mplement the RCM algorthm to survey the nfluence on the monotone teratve schemes. Fgs. 9 and show a more compact pattern produced by RCM scheme. However, the teraton numbers and CPU tmes do not be reduced for all monotone Issue 4, Volume 7, Aprl 8

8 WEA TRANACTION on COMPUTER 5 5 Error nz = Iteraton No. Fgure 9: The matrx pattern before RCM orderng. Fgure 7: The error versus the number of teratons for the case of spheres confned n a pore. old lne: Jacob method, dotted lne: Gauss-edel method and dashed lne: Pcard method. old lne wth trangles: OR method wth ω=.8, old lne wth crcles: OR method wth ω= nz = Iteraton No. Fgure : The matrx pattern after RCM orderng. schemes. The ncomplete LU factorzaton and multgrd methods should be consdered to speed up the convergence n the future. 3 5 Concluson Acceleraton Parameter (ω) An teratve method for fnte-element solutons named as the monotone teratve method s proposed for the study of the nonlnear Posson-Boltzmann equaton. Wth the help of the specal nonlnear property we can construct the maxmal sequences whch converge decreasngly to the soluton of the PB equaton. These teratve methods are globally and monotoncally convergent wth smple upper solutons of the PB equaton as ntal guesses. Pcard, Gaussedel, Jacob and OR monotone teratve methods Fgure 8: The number of teratons versus the acceleraton parameter for the case of spheres confned n a pore. IN: Issue 4, Volume 7, Aprl 8

9 WEA TRANACTION on COMPUTER are completely presented for the FE solutons. everal numercal examples are also gven and found n good agreement wth the prevous publshed results. However, t s worthy pontng out that the theoretcal analyss of convergence for the OR method n the case < ω < s stll open, and t, we hope, wll come n the near future. Acnowledgements: We would le to than the anonymous referees for the valuable comments and suggestons. We are also grateful to the Natonal Center for Hgh-performance Computng for computer tme and facltes. Ths wor was supported by NC under Grant 96-5-M-7-, Tawan. References: ] W. R. Bowen and A. O. harf, Adaptve fnte-element soluton of the nonlnear Posson- Boltzmann equaton: a charged sphercal partcle at varous dstances from a charged cylndrcal pore n a charged planar surface, J. Coll. Interface c. 87, 997, pp ] P. Dyshloveno, Adaptve mesh enrchment for the Posson-Boltzmann equaton, J. Comput. Phys. 7,, pp ] W. B. Russell, D. A. avlle and W. R. chowalter, Collodal Dsperson, Cambrdge Unversty Press, 989 4] R. E. Ban and D. J. Rose, Parameter selecton for Newton-le methods applcable to nonlnear partal dfferental equatons, IAM J. Numer. Anal. 7, 98, pp ] R.-C. Chen and J.-L. Lu, An teratve method for adaptve fnte element solutons of an energy transport model of semconductor devces, J. Comput. Phys. 89, 3, pp ] H. Amann, upersoluton, monotone teraton and stablty, J. Dff. Eq., 976, pp ]. Hela and V. Lashmantham, Monotone teratve technques for dscontnuous nonlnear dfferental equatons, Marcel Deer, New Yor, ] C. V. Pao, Nonlnear Parabolc and Ellptc Equatons, Plenum Press, New Yor, 99. 9] D. attnger, Topcs n tablty and Bfurcaton Theory, Lecture Notes n Mathematcs, vol 39, prnger-verlag, Berln-Hedelberg-New Yor, 973. ] D. Greenspan and.v. Parter, Mldly nonlnear ellptc partal dfferental equatons and ther numercal soluton II, Numer. Math. 7, 965, pp IN: ] R. Kannan and M. B. Ray, Monotone teratve methods for nonlnear equatons nvolvng nonnvertble lnear part, Numer. Math. 45, 984, pp ] C. V. Pao, Bloc monotone teratve methods for numercal solutons of nonlnear ellptc equatons, Numersche Mathemat 7, 995, pp ] K. Ishhara, Monotone explct teratons of the fnte element approxmatons for the nonlnear boundary value problem, Numer. Math. 43, 984, pp ] C. A. Brebba and. Waler, Boundary Element Technques n Engneerng, Newnes- Butterworths, London, 98. 5] Y. Deng, G. Chen, W. M. N, and J. Zhou, Boundary element monotone teraton scheme for semlnear ellptc partal dfferental equatons, Math. Comp. 65, 996, pp ] M. aahhara, An teratve boundary ntegral equaton method for mldly nonlnear ellptc partal dfferental equatons, Boundary Elements IX, C. A. Brebba and G. Maer, ed., vol. II, prnger-verlag, Berln-Hedelberg, (985) ] R.. Varga, Matrx Iteratve Analyss, Prentce- Hall, Inc. 96 8] R.-C. Chen and J.-L. Lu, Monotone teratve methods for the adaptve fnte element soluton of semconductor equatons, J. Comput. Appled Math. 59, 3, pp ]. L. Carne, D. Y. C. Chan, and J. tanovch, Computaton of forces between sphercal collodal partcles: nonlnear Posson-Boltzmann theory, J. Collod Interface c. 65, 994, pp ] P.E. Dyshloveno, Adaptve numercal method for Posson-Boltzmann equaton and ts applcaton, Comput. Phys. Commun. 47,, pp ] J. E. Ledbetter, T. L. Croxton and D. A. McQuarre, The nteracton of two charged spheres n the Posson-Boltzmann equaton, Can. J. Chem. 59, 98, pp ] W. R. Bowen and A. O. harf, Long-range electrostatc attracton between le-charge spheres n a charged pore, Nature 393, 998, pp ] A. George, Computer mplementaton of the fnte element method, Tech. Rep. TAN-C-8, tanford Unversty 97 4] Y. aad, Iteratve methods for sparse lnear systems, octy for Industral and Appled Mathematcs, 3 Issue 4, Volume 7, Aprl 8