An accurate solution of the Poisson equation by the Legendre tau method (Q1380751)

The author proposes a new method for solving the two-dimensional Poisson equation \[ \triangle u(x,y) \equiv u_{xx} + u_{yy} = f(x,y), \quad x,y \in (-1,1); \qquad u(\pm 1, y) = u(x, \pm 1) = 0. \] The method is developed using Legendre polynomials \(L_k\) to approximate \(u\) and \(f\). By orthogonalizing the residual \(R_N (u_N) = \triangle u_N - f_N\) with respect to the basis functions \(L_k\), \(N-1\) ordinary differential equations of second order are generated. An algorithm for computing the approximate solution \(X\) is also given. Numerical experiments for the test problem \(u(x,y) = \sin(4\pi x)\sin(4\pi y)\) are reported. The results are compared with corresponding outputs from two known methods for the same problem, the first one developed by \textit{D. B. Haidvogel} and \textit{T. Zang} [J. Comput. Phys. 30, 167-180 (1979; 397.65077)] and the second one, given by \textit{H. Dang-Vu} and \textit{C. Delcarte} [ibid. 104, No. 1, 211-220 (1993; Zbl 0765.65107)]. The numerical results show that the new method is more accurate for a large number of terms \(N\) in the approximate solution.