| Abstract: | The stability of time-stepping methods for parabolic differential equations is mostly a critical issue. Furthermore, solving such equations with a classical time-stepping approach can be very expensive because many small time-steps have to be taken if steep gradients occur in the solution, even if these occur only in a narrow part of the space domain. In this paper we present a discretization technique in which finite element approximations are used in time and space simultaneously for a relatively large time period called a ‘time-slab’. This technique may be repeatedly applied to obtain further parts of the solution in subsequent time-intervals. It will be shown that, with the proposed method, the solution can be computed cheaply even if it has steep gradients and that stability is automatically guaranteed. For the solution of the non-linear algebraic equations on each time-slab fast iterative methods can be used. |
