Numerical Performance of Triangle Element Approximation for Solving 2D Poisson Equations Using 4-Point EDGAOR Method
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1 International Journal of Mathematical Analysis Vol. 9, 2015, no. 54, HIKARI Ltd, Numerical Performance of Triangle Element Approximation for Solving 2D Poisson Equations Using 4-Point EDGAOR Method Mohd Kamalrulzaman Md Akhir, Jumat Sulaiman Faculty of Science and Natural Resources Universiti Malaysia Sabah Kota Kinabalu Sabah Malaysia Copyright c 2015 Mohd Kamalrulzaman Md Akhir and Jumat Sulaiman. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract This paper aims to examine the implementation of the Explicit Decoupled Group (EDG) methods. More specifically, the four-point Explicit Decoupled Group Accelerated Over-relaxation (4-EDGAOR) is investigated. Owing to the efficacy of this method, the main aim of the current research is to demonstrate the benefits of the 4-EDGAOR in solving two-dimensional (2D) Poisson equations with the help of the half-sweep triangle finite element approximation equation based on the Galerkin scheme. The results of numerical experiments demonstrate the effectiveness of the 4-EDGAOR method over the previous four point block methods (4-EDGSOR, 4-EDG, 4-EGAOR, 4-EGSOR and 4-EGGS). Based on the numerical results obtained, the results show that the 4- EDGAOR method outperforms the 4-EDGSOR, 4-EDG, 4-EGAOR, 4- EGSOR and 4-EG methods in terms of number of iterations and CPU time. Mathematics Subject Classification: 41A55, 45A05, 45B05 Keywords: Partial Differential equations (PDEs); Poisson equation, Explicit Decoupled Group (EDG), Point Block Iteration; Galerkin Scheme, Triangle element, AOR method
2 2668 Mohd Kamalrulzaman Md Akhir and Jumat Sulaiman 1 Introduction Partial Differential equations (PDEs) are implemented in mathematical models in numerous and diverse physical circumstances, as well as in re-formulations of other mathematical problems. As evident in the related literature, PDEs of the Poisson type are among the most practical and frequently investigated. Therefore, in this paper, numerical solutions of linear, 2D Poisson equations are considered. The standard form for 2D Poisson equations can be represented mathematically as follows: 2 U x 2 with the dirichlet boundary conditions + 2 U = f (x, y), (x, y) a, b a, b (1) y2 U(x, a) = g 1 (x), a x b, U(x, b) = g 2 (x), a x b, U(a, y) = g 3 (y), a x b, U(b, y) = g 4 (x), a y b, where f (x, y) is a given function with sufficient smoothness. A numerical approach to the solution of the problem (1) is a fundamental subdivision of scientific examination. Basically, 2D Poisson equations are solved numerically by discretizing the problems to the solution of linear systems. In order to obtain numerical solutions, some valid numerical methods for discretizing the problem (1) have been developed in recent years, including the mesh-free 4, 20, 21 and mesh-based methods. However, these discretization schemes mostly lead to large sparse linear systems that could involve high price in order to provide the solution, based on direct methods, with the escalation in the order of the linear systems. Hence, a substitute of these schemes could be the iterative methods that provide proficient solutions for the above-mentioned extensive problems Amongst the existing iterative methods, point block methods have been extensively accepted as efficient methods for sparse linear systems. The standard block method (also known as the 4-EGGS method 6) is a particular example of the four point block method. Apart from the standard 4-EGGS method, the variants of the four point block method, which are 4-EDGSOR methods, have also been proposed by 1. Additional research studies were executed by 9, 10, 12, to authenticate the efficacy of the half-sweep concept. Fundamentally, the EDG method is derived from a complexity reduction approach based on half- sweep concepts, respectively. To introduce the 4-EDGAOR method based on the Galerkin scheme in solving 2D Poisson equations, is the foremost objective of current research paper. As shown in Figure 1, in order to simplify the formulation of the full-sweep and half-sweep triangle element approximation equations for problem (1), uni-
3 Numerical performance of triangle element approximation 2669 (a) (b) Figure 1: (a) and (b) show the solution domain Ω of triangle elements for the full- and half-sweep cases at n = 8. form node points are discussed only. Based on Figure 1, there is a need to discretize the solution of domain evenly in both the x and y directions with a mesh size h, as given below: x = y = h = b a, m = n + 1. (2) n As depicted in Figure 1, for problem (1), the triangle finite element networks were built as a guide to derive triangle finite element approximation equations. Correspondingly, the similar concept of the half-sweep iterations was implemented in the finite difference methods 1, 16; each triangle element involved only two node points of type, as revealed in Figure 1. Thus, the full-sweep and half-sweep approaches were implemented in the same type of node points until the iterative convergence test was achieved. Afterwards, at the remaining points (points of type ), other approximate solutions were computed directly 1, 13, 16. The framework of this paper is ordered in the sequence explained subsequently. Starting with Section 2, derivation of the half-sweep triangle element approximation is elaborated at first. Followed by the Section 3 that discusses the implementation of the 4-EDGAOR method for solving problems (1). Subsequently, in Section 4, numerical results are shown to assess the performance of the examined iterative methods followed by the concluding remarks that are presented in Section 5.
4 2670 Mohd Kamalrulzaman Md Akhir and Jumat Sulaiman 2 Formulation of Half-sweep Triangle Element Approximations As discussed in the aforementioned section, to solve 2D Poisson equations, the 4-EDGAOR method was implemented by using the half-sweep finite element approximation equation based on the Galerkin scheme. The common approximation of the function U(x, y), by keeping in view only three node points of type, in the form of an interpolation function for a haphazard triangle element, e, is given by 3, 4: Ũ e (x, y) = N 1 (x, y)u 1 + N 2 (x, y)u 2 + N 3 (x, y)u 3 (3) The shape functions N k (x, y), k = 1, 2, 3, can generally be shown as: where, N k (x, y) = 1 A (a k + b k x + c k y), k = 1, 2, 3 (4) A = x 1 (y 2 y 3 ) + x 2 (y 3 y 1 ) + x 3 (y 1 y 2 ) a 1 a 2 a 3 = x 2 y 3 x 3 y 2 x 3 y 1 x 1 y 3 x 1 y 2 x 2 y 1, b 1 b 2 b 3 = y 2 y 3 y 3 y 1 y 1 y 2, c 1 c 2 c 3 = x 3 x 2 x 1 x 3 x 2 x 3 Beside this, the rst order partial derivatives of the shape functions towards x and y can be shown, respectively, as follows: (N x k (x, y)) = y (N k (x, y)) = b k det c A k det A, }, k = 1, 2, 3 (5) According to Figure 2, using the approximation of the functions U(x, y) and f(x, y), the distribution of the hat function, R r,s (x, y), in the solution domain, in the case of the full- and half-sweep for the whole domain can be defined as given below and was also described in 19: Ũ(x, y) = f(x, y) = m m R r,s (x, y)u r,s (6) r=0 s=0 m r=0 s=0 m R r,s (x, y)u r,s (7)
5 Numerical performance of triangle element approximation 2671 (a) (b) Figure 2: (a) and (b) show the definition of the hat function R i,j (x, y), of fulland half-sweep triangle elements at the solution domain. and Ũ(x, y) = f(x, y) = m m r=0,2,4 s=0,2,4 m m r=0,2,4 s=0,2,4 R r,s (x, y)u r,s + R r,s (x, y)u r,s + m 1 m 1 r=1,3,5 s=1,3,5 m 1 m 1 r=1,3,5 s=1,3,5 R r,s (x, y)u r,s (8) R r,s (x, y)u r,s (9) Hence, Eq. (8) is is an approximate solution of the problem (1). For problem (1), in order to develop the full-sweep and half-sweep finite element approximation equations, Galerkin scheme 18 was considered as follows: R i,j (x, y)e i,j (x, y) = 0, i, j = 0, 1, 2,..., m (10) D + 2 U y 2 where, E(x, y) = 2 U x 2 the Green theorem, Eq. (10) can be rewritten as: λ ( R i,j (x, y) u f(x, y) is a residual function. By incorporating y dx + R i,j(x, y) u ) x dy b b ( Ri,j (x, y) u x x + R i,j(x, y) y a a ) u dxdy = F i,j (11) y Then by simplying Eq. (10), we can derive the finite element approximation equation gives as follows: K i,j,r,su r,s = C i,j,r,sf r,s (12)
6 2672 Mohd Kamalrulzaman Md Akhir and Jumat Sulaiman where, K i,j,r,s = C i,j,r,s = b b ( Ri,j a a b b a a x ) R r,s dxdy + x ( Ri,j (x, y)r r,s (x, y) ) dxdy. b b ( Ri,j a a y ) R r,s dxdy, y As a matter of fact, for the full- and half-sweep cases, the linear system in Eq. (12) could be simply articulated in the form of stencil, as given below in Eq. (13) and Eq. (14): Full-sweep: Half-sweep: U i,j = h f i,j (13) U i,j = h2 5 1 f i,j, i = U i,j = h f i,j, i 1, n U i,j = h2 1 5 f i,j, i = n (14) 6 In fact, the stencil forms in Eqs. (13) and (14) forms consist of seven node points in formulating their approximation equations. On the other hand, two of its coefficients are zero. Apart of this, the stencil forms for both triangle finite element schemes are the same compared to the existing five points finite difference scheme, see 1, The AOR Method The following discussion can be found in 2, 5, Formulation of 4-EGAOR Method The following discussion can be found in 14.
7 Numerical performance of triangle element approximation Formulation of 4-EDGAOR Method According to Abdullah 1, in solving the 2D Poisson equation via the halfsweep finite element approximation equation, the 4-EDG method proved to be more efficient as compared to the 4-EG method. Likewise, the same steps were adopted for the finite difference approach. Let a four solid point group be selected to develop a (4x4) linear system, as shown below: where, U i,j U i+1,j+1 U i+1,j U i,j+1 = S 1 S 2 S 3 S 4 (15) S 1 = U i 1,j 1 + U i 1,j+1 + U i+1,j 1 F i,j+1, S 2 = U i+2,j + U i,j+2 + U i+2,j+2 F i+1,j+1, S 3 = U i,j 1 + U i+2,j 2 + U i+2,j+1 F i+1,j, S 4 = U i 1,j + U i 1,j+2 + U i+1,j+2 F i,j+1, and, F i,j = h2 6 (f i 2,j + f i+2,j + f i 1,j 1 + +f i 1,j+1 + f i+1,j 1 + f i+1,j+1 + 6f i,j ) The linear system in Eq. (15) can be independently decomposed into two (2x2) linear systems. Therefore, the 4-EDG method can be easily reduced as follows: u i,j u i+1,j+1 ui+1,j u i,j+1 (k+1) = (k+1) = S1 S 2 S3 S 4 (16) (17) By adding the parameter, ω into Eqs. (16) and (17), the 4-EDGSOR method can be simplified to become: u i,j u i+1,j+1 (k+1) = w S1 S 2 + (1 ω) u i,j u i+1,j+1 (k) (18) ui+1,j u i,j+1 (k+1) = w S3 S 4 ui+1,j + (1 ω) u i,j+1 (k) (19)
8 2674 Mohd Kamalrulzaman Md Akhir and Jumat Sulaiman where the value is defined in the range, of 1 ω < 2. Now, we investigate the performance of the 4-EDGAOR method which is derived on the combination between the 4-EDG and AOR methods. Therefore, by applying the AOR method 2, 12, 21 into Eqs. (18) and (19), the general scheme for this method can be shown as follows: U i,j U i+1,j+1 (k+1) = r 4T1 15 T 1 + ω S1 S 2 + (1 ω) U i,j U i+1,j+1 (k) (20) Ui+1,j where, U i,j+1 (k+1) = r 4T1 15 T 1 + ω S3 S 4 Ui+1,j + (1 ω) U i,j+1 (k) (21) T 1 = U (k+1) i 1,j 1 U (k) i 1,j 1 + U (k+1) i+1,j 1 U (k) i+1,j 1 + U (k+1) i 1,j+1 U (k) i 1,j+1, S 1 = U (k) i 1,j 1 + U (k) i 1,j+1 + U (k) i+1,j 1 F i,j+1, S 2 = U (k) i+2,j + U (k) i,j+2 + U (k) i+2,j+2 F i+1,j+1, S 3 = U (k) i,j 1 + U (k) i+2,j 2 + U (k) i+2,j+1 F i+1,j, S 4 = U (k) i 1,j + U (k) i 1,j+2 + U (k) i+1,j+2 F i,j+1. Because of the extra benefits of the AOR method, which have two weighted parameters, all of the common existing methods become unique cases of this method in the scenario the parameters take certain values. For example, when w = 1 and r = 0, we acquire the the point block Jacobi method. If w = r = 1, we acquire the point block GS method. If w = r, the point block SOR method is attained 21. Since, the coefficient matrix in Eq. (12) is a pentadiagonal matrix, it has the property A, and is Consistently Ordered 7. At this moment, to implement 4-EDGAOR method, we use Eq. (3.2) or Eq. (3.2) allows us to iterate through half of the points, lying on the 2h-grid. Again, it can be observed that Eq. (3.2) or (3.2) involves a group of points of type. To implement the iteration process, the algorithmn of the 4-EDGAOR method can be displayed as follows: 1. Discretize the solution domain into point of types (ie., ) as shown in Figure 1(b).
9 Numerical performance of triangle element approximation Perform iterations (using Eqs. (3.2) or (3.2)), taking the values of r = ω from the segment 1, 2). 3. Within the interval 0.1 from the value found in Step 2, define the optimal ω opt with a precision of 0.01 by choosing consecutive values for which k is minimal; r is taken to be equal to ω. 4. Perform experiments using the value of ω opt and choosing consecutive values of r with a precision of 0.01 within the interval 0.1 from the ω opt. 5. Define the value r opt for which k is minimal. 6. Evaluate the solutions at the remaining point of type using Eq. (13). 7. Display approximate solutions. 4 Numerical Results In order to compare the recitals of the methods described in the previous sections, several experiments were carried out on the following 2D Poisson example 11: 2 U x U y 2 = (cos(x + y) + cos(x y)). (22) and the exact solution is given by U(x, 0) = cos x, U ( x, π 2 ) = 0, U(0, y) = cos y, U(π, y) = cos y. U(x, y) = cos(x) cos(y). The numerical experiments were carried out on a dedicated personal with an PC Intel(R) Core (TM) i7 CPU 860@3.00Ghz, and 6.00GB RAM. The programming codes were written in C++ programming language. The AOR method obtained in this paper is compared to other methods (4-EDGSOR, 4-EDG, 4-EGAOR, 4-EGSOR and 4-EG). The value of the initial iteration is set to be zero for the test problems and in the course of implementation the tolerance error, is considered ε = For convenience, there are three vital parameters to be measured, including the number of iterations (k), maximum absolute error (Abs.Error) and the execution time (t in seconds). The numerical results of the experiment for the proposed iterative methods are given in Table 1.
10 2676 Mohd Kamalrulzaman Md Akhir and Jumat Sulaiman Table 1: Comparison of the number of iterations, execution time (seconds) and maximum absolute error for the iterative methods. n Methods r w k t Abs.Error 4-EG e-7 4-EDG e-6 4-EGSOR e EDGSOR e-6 4-EGAOR e-7 4-EDGAOR e-6 4-EG e-7 4-EDG e-6 4-EGSOR e EDGSOR e-6 4-EGAOR e-7 4-EDGAOR e-6 4-EG e-7 4-EDG e-6 4-EGSOR e EDGSOR e-6 4-EGAOR e-7 4-EDGAOR e-6 4-EG e-7 4-EDG e-6 4-EGSOR e EDGSOR e-6 4-EGAOR e-7 4-EDGAOR e-6
11 Numerical performance of triangle element approximation Conclusions In this paper, we have presented an application of the 4-EDGAOR method for solving sparse linear systems generated from the discretization of the 2D Poisson equation equations by using the Galerkin scheme. The numerical results obtained for the proposed problem (Table 1) clearly show that applying the AOR methods reduces the number of iterations, and execution time, compared to the SOR and GS methods. At the same time, it has been shown that applying the half -sweep approach reduces the computational time in the implementation of the iterative method. Overall, the numerical results demonstrate that the 4-EDGAOR method outperforms the existing block methods (4-EDGSOR, 4-EDG, 4-EGAOR, 4- EGSOR and 4-EG), particularly in the sense of the number of iterations and execution time. This is mainly attributable to the reduction of the computational complexity; since the implementations of the 4-EDGAOR method only consider approximately half of all interior node points in a solution domain. For future work, the capability of the quarter-sweep approach 2, 12, 21 should be investigated in terms of the point block iterative method by using the Galerkin scheme. References 1 A.R. Abdullah, The Four Point Explicit Decoupled Group (EDG) Method: A Fast Poisson Solver, Intern. J. of Comp. Math., 38 (1991), A. Hadjidimos, Accelerated Over Relaxation method, Math. of Comput., 32 (1978), C.A.J. Fletcher, The Galerkin method: An introduction. In: Noye, J. (pnyt.), Num. Simul. of Flu. Mot., Amsterdam, North-Holland Publishing Company, 52 (1978), C.A.J. Fletcher, Computational Galerkin Method. Springer Series in Computational Physics, Springer-Verlag, New York, D. J. Evans and M.M. Martins, The AOR method For AX - XB = C, Int. J. of Comp. Math., 52 (1994), D. J. Evans, Group Explicit Methods for the Numerical Solutions of Partial Differential Equations, Gordon and Breach Science Publishers, Australia, 1997.
12 2678 Mohd Kamalrulzaman Md Akhir and Jumat Sulaiman 7 D.M. Young, Iterative Solution of Large Linear Systems, Academic Press, London, G. Yagawa and T. Furukawa, Recent developments of free mesh method, Int. J. for Numer. Meth. in Engi., 47 (2000), J. Sulaiman, M.K. Hasan and M. Othman, Red-Black Half-Sweep Iterative Method Using Triangle Finite Element Approximation for 2D Poisson Equations, Lecture Notes in Computer Science, 4487 (2007), J. Sulaiman, M.K. Hasan, M. Othman, Red-Black EDGSOR Iterative Method Using Triangle Element Approximation for 2D Poisson Equations, Lecture Notes in Computer Science, 4707 (2007), M.K.M. Akhir, J. Sulaiman, Triangle Element Analysis for the Solutions of 2D Poisson Equations via AOR method, J. of Adv. in Math., 11 (2015), no. 2, M.K.M. Akhir, J. Sulaiman, HSAOR Iterative Method for the Finite Element Solution of 2D Poisson Equations, Int. J. of Math. And Comp., 27 (2016), no. 2, M.K.M. Akhir., J. Sulaiman, The 4-EGAOR Method for Solving Triangle Element Approximation of 2D Poisson Equations, App. Math. Scie., 9 (2015), M.K.M. Akhir, J. Sulaiman, Analysis of Triangle Element Approximation fopr Solving 2D Poisson Equations using QSAOR, Global Journal of Pure and App. Math., 2015, In Press. 15 O.C. Zienkiewicz, Why finite elements?, Finite Elements in Fluids Volume, ln. R.H. Gallagher, J.T. Oden, C.Taylor, O.C. Zienkiewicz, (Eds.), John Wiley and Sons, London, P.E. Lewis and J.P. Ward, The Finite Element Method: Principles and Applications, Addison-Wesley Publishing Company, Wokingham, R. Vichnevetsky, Computer Methods for Partial Differential Equations, Vol I, Prentice-Hall, New Jersey, 1981.
13 Numerical performance of triangle element approximation T. Belytschko, Y. Krongauz, D. Organ, M. Fleming, and P. Krysl, Meshless methods: An overview and recent developments, Comp. Meth. in App. Mech. and Eng., 139 (1996), T. Zhu, A New Meshless Regular Local Boundary Integral Equation (MRLBIE) approach, Int. Journal for Num. Meth.s in Eng., 46 (1999), W.S. Yousif and D.J. Evans, Explicit Decoupled Group Iterative Methods and Their Implementations, Parallel Algorithms and Applications, 7 (1995), W.S. Yousif and M.M. Martins, Explicit De-couple Group AOR method for solving elliptic partial differential equations, Neural, Parallel and Scientific Computations, 16 (2008), no. 4, Received: October 4, 2015; Published: December 2, 2015
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