Jacobi and Gauss Seidel Methods To Solve Elliptic Partial Differential Equations

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1 Jacobi and Gauss Seidel Methods To Solve Elliptic Partial Differential Equations Mohamed Mohamed Elgezzon Higher Institute for Engineering Technology-- Zliten Abstract Solving elliptic partial differential equations manually takes long time. Computer plays an important role to solve this kind of equations. Finite element method is used to solve elliptic partial differential equations by reducing these equations to linear system Ax = b using iterative methods as Jacobi and Gauss Seidel methods. These methods are applied for two examples in this paper. Keywords: iterative methods elliptic partial differential equation finite element method. الملخص: حم ان عادالت انتفاضهية انجزئية اننالصة يذويا تأخذ ولتا طىيل حيث ا انحاسىب )انك ثيىتز( يهعة دورا يه ا نحم هذا اننىع ي ان عادالت وتستخذو طزيمة انعنصز ان حذد نحم ان عادالت انتفاضهية انجزئية اننالصة تىاسطة تخفيض هذه ان عادالت إنى طزق اننظاو انخطي Ax = b تىاسطة استخذاو طزق انتكزار يثم طزيمة جاكىتي 1

2 وطزيمة جاوس. و في هذه انىرلة انثحثية يتى تطثيك انطزيمتي ي خلل يثاني نذراسته ا في هذا انثحث. Introduction. Let be a bounded domain in R d, with boundary. It is assumed that f and g are continuous on. So a unique solution exists - = f(x,y) For(x,y) (1) u(x,y) = g(x,y) in = {(x,y) : a < x < b,c < y < d}. The concept of the finite element is to apply this to equation (1). First, the test function v is picked where v satisfies the boundary condition v = 0 on and multiply the first equation by v. Finally, we integrate the equation over, using Green's formula as = - v ds +. A finite element discretization of (1) is based on the weak formulation:[2] seek u V = [H ( )] d such that. A(u, v) = (f,v) V (2) Where a(u, v) =, (f,v) = The approximate solution u h V h V satisfies 2

3 a(u h, v) = (f,v) V h (3) we define u h = following equations.[3] u i i (x), thus, we substitute u h in the a(u h, k) = (f, k) k = 1,, N We obtain u i.a( I, k) = (f, k) k = 1,, N (*) We could write the formula in (*) as: Ax = b. (4) Where A R n n is symmetric matrix, b R n, and x R n is the request. To solve (4) we use the iterative technique as follows solution[4] X k+1 = Tx k + C, k= 0,1,.., (5) Where x 0 is given For the convergence it is required that for all x 0 R n the vector sequence produced by the iteration (5) should be convergent to the same vector x * R n. And x * would be approximate solution of x in (4), that is, Ax * = b. The following two theorems are needed for the seek of completeness.[5] 3

4 Theorem 1. Assume that (M,N) is a splitting of the regular matrix A, such that T = M -1 N is convergent matrix. Then the iterative method (5) based on this splitting is convergent to the solution of the system of linear algebraic equations Ax = b. Theorem 2. Assume that A is monotone matrix, Then any regular splitting of A generates a convergent iterative method, that is, if (M,N) is a regular splitting of A then p(m -1 N) < 1. In the following, some well-known iterative methods are considered to the solution of (4) such as, Jacobi method and Gauss Seidel method. (1) Jacobi iterative method (JIM). The Jacobi iteration of the form: X = b i - a ij X i = 1,2,,n, k= 0,1,2,.. (6) Is considered, where it is assumed that a ii notations[6] 0. Introducing the D = diag A; -U = upper triangular part of A; -L = lower triangular part of A; We have the obvious relation A = D U L. 4

5 Then the Jacobi iterative method (6) can be rewritten in the matrix form X k+1 = D -1 (b + (U + L) x k ), )7 ( That is, X k+1 = D -1 (U+L)x k + D -b b. Obviously, this is a BIM with G J = D -1 (U+L) and c J = D -1 b. On the other hand, Dx k+1 = (U+L) X K + b. That is, if M J = D and N J = U+L is defined then M J x k+1 = N J x k + b. Moreover, the relation M J N J = D U L = A holds. Consequently, the iterative method of the form (7) is BIM based on the splitting A = M J N J. (2) Gauss Seidel iterative method (GSIM). The Gauss Seidel iteration of the form Xi k+1 = b i - a ij x Xj k+1 - a ij x Xj k i= 1,2,.n, k= 0,1,2. (8) The Gauss Seidel method can be rewritten in the matrix form 5 Dx k+1 = Lx k+1 + Ux k + b.

6 It can be remarked that the main difference between the Gauss- Seidel method and the Jacobi method is the following:[7][8] Using the Jacobi method for the computation of the new iteration the previous approximations are applied in each equation, while in Gauss Seidel the new already computer in more suitable form is rewritten as (D - L)x k+1 = Ux k + b. (9) Introducing the notations M GS = D L, N GS = U, We obtain M GS N GS = D L U = A. That is, M GS N GS is a splitting of A. Then (9) can be rewritten in the form M GS X K+1 = N GS X K + b; or X k+1 = T GS X K + C GS; Where, we have used the notations T GS = M GS -1 N GS = (D-L) -1 U, C GS = M GS -1 b = (D-L) -1 b. Example 1 Consider the follows problem with boundary conditions and given domain - = 2 (x 2 + y 2 ) 0 x,y 1 6

7 u(x,y) = 0 when x= 1, y = 1; u(x,y) = (1-y 2 ) u(x,y) = (1-x 2 ) when x=0; when y=0; Example 2 Consider the follows problem with boundary conditions and given domain - = 2 (x 2 + y 2 ) 0 x,y 1 u(x,y) = 0 in boundary Briefly the solution of the given problem by theoretical methods is applied as defining step size h and partitioning the intervals [a,b] and [c,d] into n equal parts and the finite element method.[9] 7

8 Figure 1. The shape of example 1 where n=25 Figure 2. The final shape of matrix A of example 1 where n= 10 8

9 For example 1 Ax = b when n = 2 A=, b = [ ] [ ] x = [ ] 9

10 Figure 3. The shape of example 2 where = 25 Figure.4. The final shape of matrix A of example 2 where n = 20 11

11 For example 2 Ax = b when n = 2 =, b = [ ] [ ] x = [ ] The results of numerical tests of the Jacobi an gauss seidel iterative methods for example 1 and example 2 are given in the following tables: Table 1 Iteration and residual of matrix A by Jacobi and Gauss Seidel method of example 1 h 1 /5 1 /10 1 /20 Iteration Residual Iteration Residual Iteration Residual JIM e e e-4 GSIM e e e-4 11

12 Table 2 Iteration and residual of matrix A by Jacobi and Gauss Seidel method of example 2 h 1 /5 1 /10 1 /20 Iteration Residual Iteration Residual Iteration Residual JIM e e e-4 GSIM e e e-4 References [1]. Aleyan O. A.: On matrix splitting and application into iterative methods for linear systems An Academic. Intellectual, Cultural, Comprehensive and Arbitrated Journal Issued by the Faculty of Arts and Sciences in Zliten, Al- Merqab. Issue No (JUNE) [2]. Berman, A. and plemmons, R. Nonnegative matrices in the mathematical sciences, Academic press, New York, [3]. Hom, R.A. and Johnson, C.R. Matrix analysis, Cambridge University press, Cambridge, [4]. Ortega, J.M. and Rheinboldt, W.C. Iterative solution of linear systems, plenum press, New York, [5]. Miroslav Fiedler., Special matrices and their applications in numerical mathematics, Kluwer, [6]. Miroslav Fiedler and Vlastimil ptak, On matrices with non- positive off- diagonal elements and positive principal minors, Czechosl. Math. Journal (87) 12, 1962, pp

13 [7]. Varga, R.S. Matrix iterative analysis, prentice Hall, New York, [8]. Young, D. Iterative solution of large linear systems, Academic press, New York, [9]. Richard L. Burden and J. Donglas Faires Numerical analysis, third edition prindle, weber & Schmidt, Boston,