Approximation of numerical solution of parabolic problems

To solve a parabolic initial-boundary value problem we apply a space-time finite element method to the variational formulation. After a subdivision of the interval [0,T] in subintervals, a discretization in each slab is performed, taking both the approximate solution and the test functions as linear combinations of piecewise linear functions. We obtain in this way an iterative process which allows to get the solution at the time $t_{n+1}$ when the solution time $t_n$ is known. On such values, extrapolation methods such as Aitken's $\delta^2$ process and the epsilon-algorithm are used to predict the solution at the time T and to compute the approximate solution of the steady state.