Abstract
The finite element collocation method is sometimes equivalent to the Galerkin's method. The two methods are therefore considered together her. First step in the Ordinary Differential Equation with two examples and comparing with the analytical solution. The Galerkin method applied also in variation form. Second step in the partial differential equation with two examples; the first example using "Homogeneous heat conduction equation", and the second example using "The non-homogeneous heat conduction equation with non-homogeneous boundary condition. First the equation is transformed into a variation form by continuous time Galerkin's method and expressed in terms of linear spline basis functions. On the other hand, the collocation method is applied on the heat equation with cubic splines as the basis function. The two methods of discrimination transform the heat equation to the system of ordinary differential equations in time, respectively. The corresponding stiffness matrices for the two systems of equations turn out to be the same. These matrices are sparse, banded and symmetric, and therefore convenient for computer application