In the previous chapters we have considered fully discrete schemes for the heat equation which were derived by first discretizing in the space variables by means of a Galerkin finite element method, which results in a system of ordinary differential equations with respect to time, and then applying a finite difference type time stepping method to this system to define a fully discrete solution. In this chapter, we shall apply the Galerkin method also in the time variable and thus define and analyze a method which treats the time and space variables similarly. The approximate solution will be sought as a piecewise polynomial function in
of degree at most
– 1, which is not necessarily continuous at the nodes of the defining partition.