On direct methods for the discretization of a heat-conduction equation using spline functions

Abstract

The finite element collocation method is sometimes equivalent to the Galerkin method. The two methods are therefore considered together here. The nonhomogeneous heat-conduction equation with nonhomogeneous boundary condition is chosen for consideration. The equation is transformed into a variational form by continuous time Galerkin method and expressed in terms of linear spline basis function. On the other hand, the collocation method is applied on the heat equation with cubic splines as the basis functions. The two methods of discretization transform the heat equation to systems 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 are therefore convenient for computer application.