AN ACCURATE SOLUTION OF THE POISSON EQUATION BY THE LEGENDRE TAU METHOD

. A new Tau method is presented for the two dimensional Poisson equation Comparison of the results for the test problem u(z, 3/) sin(47rz)sin(47ry) with those computed by Haidvogel and Zang, using the matrix diagonalization method, and Dang-Vu and Delcarte, using the Chebyshev collocation method, indicates that our method would be more accurate


INTRODUCTION
Haidvogel and Zang [1] developed a matrix diagonalization method for the solution of the two- dimensional Poisson equation.This method is efficient but requires a preprocessing calculation of the eigenvalues and eigenvectors which limits the accuracy of the solution to that of the preprocessing calculations, especially at large N values Dang-Vu and Delcarte [2] developed a Chebyshev collocation method for solving the same problem Their method has the same accuracy as the matrix diagonalization method when N is small and it is more accurate when N is large In this paper we present a new alternative method for solving Au(x,y) =u=z +u= f(x,y), x,y (-1,1) which is more accurate than the above two methods

PRELIMINARIES
In this section we give a basic definition and some facts which we use hereafter DEFINITION 1.The Legendre polynomial {Lk(x), k 0, 1 are the eigenfunctions of the singular Sturm-Liouville problem Like other orthogonal polynomials the Legendre polynomials satisfy many relationships perhaps the most basic one is the orthogonality relation L,(x)Lm(x)dx (n -[" 0.5)-lnm  ( 1) [p(p+l)-n(n+l)]f, S)= ,+ and A (n + 0.5) f(x)L,(x)dx.
(2 5) 2n-1 2n+3 For more details, see Schwarz [3] 3. LEGENDRE-TAU METHOD FOR SOLVING TWO-DIMENSIONAL POISSON EQUATION The basic topics of this section involve the Legendre-Tau method to discretize a class of linear boundary value problems of the form of Problem (1 1).To explain the total procedure both analytic and numerical results are presented Referring to the boundary value problem (1.1) approximate-u and f in terms of Legendre polynomials as N ug(x,y) E a,(x)L.(y),For the approximate solution UN, the residual is given by Rz(,) zX,z(, u)-f(, u). (31) Thus, the residual can be written as N k=0 where a(k 2) is given in (2.4) and a(x) is the second derivative of ak(X) with respect to x As in a typical Galerkin scheme we generate (N 1) second order ordinary differential equations by orthogonalizing the residual with respect to the basis functions Lk (y)   (RN,Lk(y)) RgLk(y)dy 0, for k 0" N-2.
TItEOREM 3.1.The matrix G I + q.Tq+ + q_Tq_ is a nonsingular matrix PROOF.Let A be any eigenvalue of the matrix G associated with the eigenvector x such that Then, the smallest eigenvalue of G is at least 1, which implies that G is nonsingular matrix Now, multiply both sides of the differential equation (3 10) by Q (I + qT+q+ + qT_q_ )-1 to get It is easy to see that Q (I-aqT+q+)(I-qT_q_(I-aqTq+)), 1 1 cz I + q+ qT+ I + q_ (I qT.q+ )qT_ For more details, see Hager [4] Since each component of Q(I + D2C[)R is a polynomial of degree at most N/2, so we will approximate the solution of equation (3.11) by t--0 where S D2Q(CT + Ce) + D'QC[C and N/4] is the largest integer less than or equal N/4 Let H be the transition matrix from the basis ,1-{Lo(x),LI(x),...,LIN/4].I} to the basis 2 {1,x, ...,xIN/4]+} for the space P[N/4|+I {f: f is a polynomial of degree _< N/4] / 1} with usual addition and scaler multiplication Let F be the matrix of the differential operator D :P|g/4|+l P|lv/4]-i using the standard basis.Thus, the algorithm for computing X is given as follows ALGORITHM 3.1.

NUMERICAL RESULT
In this section, we give two experimental examples to show how Algorithm (3.1) works nicely Also, comparison of the results for the test problem u(x, y) sin(47rx)sin(47ry) with those computed by Haidvogel and Zang, using the matrix diagonalization method, and Dang-Vu and Delcarte, using the Chebyshev collocation method will be done (3 ll)  All the calculations are realized using the 486 IBM computer Programs are written in double precision EXAMPLE 4.1.
The exact solution is u(x, y) sin(47rx)sin(47ry) We will study the relation between the number of terms in the approximation solution N and the error in the approximation e N This relation is given in Table (1) where f(x,y) 32[(x x)(x + y2 1) + (y2 y)(y + x 1)]e2x+2u-2 The exact solution is u(x, y) 16(x x)(y y)e 2-2u-2 We will study the relation between the number of terms in the approximation solution N and the error in the approximation eN This relation is given in Table (2) In this case, first we will use the following transformation to the square [0,1] [0,1] into the square [-1,1] [-1, 1] z=2x-1, w=2y-1.From Table (1) and Table (2), we see that our method is an accurate method Compared with the Haidvogel-Zang method and Dang-Delcarte method our method should generate more accurate results at large N values 1 and the endpoint relation L,(=kl)-(:J=l)n.

Table ( 2
) Maximum Absolute Error e as a function of N N