A fast Galerkin-spectral method based on discrete Legendre polynomials for solving parabolic differential equation

The goal of this investigation is to achieve the numerical solution of a two-dimensional parabolic partial differential equation(PDE). The proposed method of this paper is based on the discrete Legendre Galerkin method and spectral collocation method to simplify the spatial derivatives and time derivatives. The discrete Galerkin method is a very fast technique compared to the classical Galerkin method since a finite sum is needed for determining the interpolation coefficients. The operational matrix of the discrete Legendre polynomials is introduced to discretize the time derivatives. Using these couple of techniques and the collocation method, the aforementioned problem is transformed into a solvable algebraic system. The results of applying this procedure to the studied cases show the high accuracy and efficiency of the new method.