The method of fundamental solutions for solving Poisson problems

Transcription

1 The method of fundamental solutions for solving Poisson problems C.J.S. Alves', C.S. Chen2and B. %der3 'Centro de Matematica Aplicada, Instituto Superior Thaico,Portugal 2Departmentof Mathematical Sciences, University of Nevada, UXA 3LaboratoryforMultiphase Processes, Nova Gorica Polytechnic, Slovenia Abstract Traditionally t,he method of fundamental solutions (MFS) is used to approximate solution of linear homogeneous equations. For nonhomogeneous problems, one needs to couple ot,her numerical schemes, such as domain integration. polynomial or radial basis functions interpolation, to evaluate particular solutions. In this paper we propose to unify t,hemfs as a numeri- cal method for directly approximating homogeneous solution and part,icular solut'ion in a similar manner. The major advantage of such approach is that the particular solution can be easily obtained and evaluated. The numerical results show that such approach can be highly accurate. 1 Introduction The method of fundamental soht~ions (MFS) was initially introduced by Kupradze and Aleksidze [ll]and has been furt,her developed by numerous mathematicians and scientists over the past 35 years. However, the method was essentially restricted to solving homogeneous elliptic equations, such as the Laplace and the biharmonic equat,ions [7,8 1. Since 1993 Golberg and Chen [8]have extend the MFS using radial basis functions (RBFs) to solving linear and non-linear Poisson problems, inhomogenous Helmholtz equations and time dependent PDEs. The key concept of such extension is

2 68 Bou~iw-~~ Elmcnts XXIV based on the evaluation of particular solution by using the dual reciprocity method (DRM) [14]and radial basis functions (RBFs) [8]. The MFS and DRM approach have achieved mesh-free method and becomes very popular in the BEM community. The main difficulty of such scheme is t,heneed of deriving an approximate analytical particular solutions which often become very complicated to evaluate [ 121. The primary goal of this paper is to directly extend t,he MFS to solving nonhomogeneous problems without using the RBFs so that an easy evaluation of particular solution can be achieved. This is apparently the first time that the MFS is used directly (without RBF's) to solve a nonhomogeneous PDE. As in [l],in this paper, we will focus on solving Poisson's equation. Many numerical schemes for solving partial differential equations (PDEs) rely on effective met,hodsof approximation techniques. In this paper we develop an interpolation scheme using the MFS. As a result, the MFS can be applied to intepolate functions and solve homogeneous PDEs simultaneously. Combining these two special features, a nonhomogeneous PDEs can be solved without t,he need of using radial basis functions or other interpo- &ion schemes. 2 The MFS for Laplace equation Consider the Laplace equation wkh Dirichlet boundary condkion Aw(P)= 0, F E R, w(p)= g(p), F E m where R c R",n = 2,3. The basic idea (1) (2) of the MFS is to approximate t,he solution W of equations (1)-(2) by expressed as a linear combination of fundamental sohhions [3] where G(P,Q)=-log / I P- &/l /27r is the fundamental solution of the operator. and {QJ};" = {(zj,yj)}fr are M distinct points on the fictitious boundary D of f2 as shown in Figure 1 for the 2D case. In general, the points { Q J } ~are chosen a priori in a certain fixed setting. Further details can be explained by the approximation analyses given by Bogomolny [3]and Cheng's convergence results for Laplace's equation with Dirishlet boundary condition when D and D are concentric circles [S]. In the numerical implement,ation,we have proven in our computations that the choice of placing the source points {Q,-}yequally spaced around a circle of radius R in R'or equally spaced in the polar coordinates (q5, Q) on a sphere of radius R in R3 gives excellent numerical results.

3 B o u d u r - yekww~tsxxiv 69 Collocation points on the boundary (MESHLESS) Source points on the source-set Figure 1. The MFS for the homogeneous solution. Once the source points have been chosen and fixed; {aj}: U {Q} in (3) are normally obtaincd from simple collocation. By choosing M + 1 points {Pk}yfl on D and M points {QJ}? on the fictitious boundary D that satisfy the equations (for Dirischlet boundary condition), we obtain M CaJG(P~,QJ)+ao=g(P~): l<k<m+l. (4) j=l From equation (4) it can bc observed that the system will becomc highly ill-conditioned as the value of R increases [4,?, 151. It is natural to put a limit on the value of R to approximately 5 10 times of the diameter of D. Despite the ill-conditioning problem, the accuracyof the numerical solution is generally insensitive to the large value of R Surface interpolationusing the MFS To extend the MFS to nonhomogeneous problems, we consider the following Poisson's equation Let upbe a particular solution of (5) 4 P )=.f(p), p E R, (5),(P) = g p ), P E ar. (6) 4l,(P)=W? (7) but does not necessarily satisfy the boundary condition (6). Let U = U-up. Then, v satisfies fqp) = 0, P E (1, (8).(P) = g(p)- Up(P), P E m. (9) Notice that thc approximate solution d of 'U can be obtained as described in the last section provided that up can bc successfully evaluatcd. Since

4 70 Bou~iw-~~ Elmcnts XXIV up is not unique, it. allows a wide variety of choices for finding a suitable particular solut,ion. In fact,, in recent years, the evaluation of approximate particular solution of upis a subject of intensive research. Traditionally up was evaluated by Newtonian potent,ial which required tedious domain integration. In his paper, Atkinson [2]gave three different ways to evaluate upin (7). A review for the particular solut,ionwas given by Golberg [lo]. In the engineering literature: the DRM 1141 and the MRM [13]are mostly widely used to directly approximate the par- t~icular solut,ion. In particular, RBFs have been widely used in this respect in the DRM beratwe. One of the most, popular choices of t,he RBFs is the thin plate splines (TPS) [g] In the DRM, we approximate f(p)in (7) by a linear combination of basis function {pj};; Le., f(p)cx f(p)=2clj$cqp) j=1 where {ai}? are undet,ermined coefficients. By collocation. where {P,}';are n points in Rd.Least-squares method can also be usedif the number of collocat,ion and source points are not equal. Notice that since the particular solution does not have to satisfy the boundary condition, the interpolation points {P,}: can be lied out'side the domain. For the traditional DRM, {&}F are chosen inside the domain. Assuming that (13) can be solved uniquely for {aj}:, the approximate particular solution Q, is given by n where AQJ = pj, 1L j 5 n. 05) To achieve high efficiency. it is important to solve (15) analytically. For Laplacian A, the closed-form QJis not' difficult to obtain. However, for other differential operators, the derivation of is not trivial. A great deal of

5 A) effort in deriving the closed-form particular solution for various differential operators [8]. We have observed that the TPS p(.) in (11) is a fundamental solution of the biharmonic equation A2p= -6, (16) where S is the Dirac delta. Motivate by this observation: we seek an approximation based on fundamental solution paof the following eigenvalue equation, which is also a modified Helmholtz equation, (A ~ px= -6 (17) where A E R+\(0) and Similar to the TPS interpolation in (la), an approximation of f can be written as the linear combination of fundamental solutions which contains different wave lengths; i.e., f(p)= 5ii:ai!dPA,k! i=l k l where q is the number of wave length. Note that 'pa has a singularity at T = 0. To avoid the singularity using interpolation scheme, we choose the source points outside the domain as shown in Figure 2. One major advantage of choosing {yxi} as basis function is that the particular solution cm beobt,ained directly. It is now clear that if the approximate particular solution I& is written in the form W? =i:f ) 2 A i k > i=l k=l then by Q n q n i=l k l i=l k=l This implics that if WC choose pik= aik/at in (19), then 6,in (20)is indeed an approximate particular solution. In contrast to the RBF approach, it is intcrcst to observe that the basis functions of particular solution in (21) and interpolation in (20) are the same. In numerical implementation, we use

6 77r WIT Press, Ashurst Lodge, Southampton, SO40 7AA, UK. All rights reserved. ISBN BOLM~~I-J~ Elcmcnts XXIV least-squares method insteadof collocation method. More collocation points in the domain t,han source points on the fictitious boundary are obviously needed. Figure 2. The MFS for surface interpolation. Figure 2. The MFS for surface interpolation. To justify the approximationof f in (20),we need the following theorems 111. Theorem l Let {Q}: $ 0. Th,e fmctions ~PA,(P,QI):... >c~x,(p,qn);... ;~PA,(P.QI),...,PA,(P,Q.~) restricted to any open set R are linear indepe,n&nt. Theorem 2 Let f' be an admissible source set and I an intemdin (-m, 01. The space is dense in L2(L?). 4 Numerical results +,I$ = SW{PA(P, & ) I n : Q E i., x E I} To demonstrate the effectiveness of our proposed approach, we consider a benchmark example [6]for Poisson's equation (5)-(6) where 7517r2sin r"c sin 7nz sin 3?ry sin 57rY ~ cos cos 2 f(xly1 = ry h y 157r rx 37ry 57ry X sin-sin-+-sm-an-cos-cos-, g(2,y) zz sin sin sin sin?e, 4 and R U dr = [l,21 X [l,21. The graph of nonhomogeneous term f(x,y) is shown in Figure 3. The exact solution u(z,y) is given by 2711:. 7nx. 3ny 5ny u(z,y) = sin7sm4sm sin T.

7 WIT Press, Ashurst Lodge, Southampton, SO40 7AA, UK. All rights reserved. B o u d u r - yekww~tsxxiv 73 Figure 3. Profile of the forcing term f. To interpolate the forcing term f(z,y), we choose 400 quasi-random interior points and 40 uniformly distributed points on the fictitious boundary which is a circle with center at (1.5> 1.5) and radius5. The wave numbers in (20) are choosing to be {l,10,20,...,so}. Least square method has been used for this purpose. We use least square subroutine DLSQRR from IMSL library. To evaluate the approximate homogeneous solution 6 of (8)-(g), we choose 40 uniformly distributed points on the physical boundary and the same numbcr of points on the fictitious boundary same as above. Colloca- tion method has been used to evaluate 6.We perform our numerical test on 400 quasi-random points in the domain. Thc maximum absolute error of U is X lo-'. The maximum absolute interpolation error of f is X 1O-l'. The overall profile of absolute error of U and f are shown in Figures 4 and 5. Table 1. The effect of U and f for various number of wave lengths. 1 #ofwavclcngth 1 ~ l u - C ~ l o o 1 1 l. f - f l l l l llro E ~ E E E ~ 10 In Table 1,we show the maximum absolute errors of U and f for various wavc lcngths with 40 fixed source points on the fictitious boundary. This reveals that the number of wave length has great impact on the numerical solution of U. Double precision is used for all the numerical computation

8 ' 2002 WIT Press, Ashurst Lodge, Southampton, SO40 7AA, UK. All rights reserved. 74 Bou~iw-~~ Elmcnts XXIV 2 2 Figure 4. Profile of absolute error of U. x 1o:'O 1 2' 2 Figure 5. Profile of absolute interpolation error of f. 5 Conclusions In the past the MFS has been used to solve the homogeneous equation. In this paper we propose to use the MFS to interpolate the forcing term

9 B o u d u r - yekww~tsxxiv 75 directly. As a result, the particular solution of the nonhomogeneous solution can be evaluated easily. The preliminary numerical results shows that our approach is robust and compare favorable to the RBF approach. There are stilla number of issues such as the numberof wave numbers, the number and the location of source points needed to be further studied. The extension of this work to more general differential equations such as nonlinear and timc-dependent problems is current,ly under investigation. Acknowledgement: The work is supported by a NATO grant under reference PST.CLG The first author s work is also partially supported by project FCT-POCTI/MAT/34735/99. References [l]c.j.s. Alves and C.S. Chen, The MFS method adapted for a non homogeneous equation, ICES 01 Conference Proceeding in CD ROM. [2] K.E. Atkinson, The numerical evaluation of particular solutions for Poisson s equation, IMA Num. Anal., 5,pp , [3] A. Bogomolny, Fundamental Solutions Method for Elliptic Boundary Value Problems, SIAM J. Numer. Anal.. 22, pp , [4] K. Balakrishnan and P.A. Ramachandran, The method of fundamenhl solutions for linear diffusion-reaction equations, Math. comput. Modelling, 31, pp , [5] R.S.C. Cheng, Delta-Trigonometric and Spline Methods using the Single-Layer Potential Representation, Ph.D. dissertat.ion, University of Maryland, [6] AXD. Cheng, 0.Lafe, and S. Grilli, Dual reciprocity BEM based on global interpolation functions, Eng. Andy. Boundary Elements, 13, pp , [7] G.Fairweather and A. Karageorghis, The method of fundamental solutions for elliptic boundary value problems, Adv. Comp. Math., 9, pp , [B] M.A. Golberg and C.S. Chen, The method of fundamental solutions for potential, Helmholtz and diffusion problems, Chapater 4, Boundary Int egral Methods: Numerical and Mathematical Aspects, ed. M.A. Golberg, WIT Press 8~Computational Mechanics Publications, Boston, Southampton, pp , [g] M.A. Golberg and C.S. Chen, The theory of radial basis functions applied to t,he BEN1 for inhomogeneous partial differential equations, Boundary Elements Communications. 5: pp , 1994.

10 76 Bou~iw-~~ Elmcnts XXIV [lo] M A. Golberg, The numerical evaluation of paticular solutions in the BENl - A review, Boundary Elements Communications. 6, pp , ill] V.D. Kupradze and M.A. Aleksidze, The method of functional equapp , tions for the approximate solution of certain boundary value problems, U.S.S.R. Comput,ational Mathematics and Mathematical Physics, 4, A.S. Muleshkov, MA. Golberg & C.S. Chen, Particular solutions of Helmholtz-type operators using higher order polyharmonic splines, Computational Mechanics, 23, pp , [13] A.J. Nowak and A.C. Neves, Fundament'als of the multiple reciprocity met,hod. in: The Multiple Reciprocity Boundary Element Method, editors: A.J. Nowak and A.C. Neves. Computational Mechanics Publica- tions, Southampton, pp , [14]P.W. Partridge, C.A. Brebbia and L.C. Wrobel, The Dual Reciprocity Boundary Element Method, Comput,at,ional Mechanics Publications [15] Y-S. Smyrlis and A. Karageorghis: Some aspects of the method of funpp. damental solutions for certain harmonic problems, J. Sci. Comput., 16, ,2001.