Calculation of Transient Temperature Fields with Finite Elements in Unstructured Space and Time Dimensions.

Abstract

The stability of time-stepping methods for parabolic differential equations is 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 they occur only in a small 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. The weighted residual process is used to formulate a finite element method for a space-time domain based upon the continuous Galerkin method. This technique may be repeatedly applied to obtain further parts of the solution in subsequent time intervals.