Galerkin approximation of the time derivative in the finite element analysis of groundwater flow

TLDR: The Galerkin method of approximation permits a high-order approximation in time as well as in space as mentioned in this paper, and the resulting approximate equations have been successfully solved by using a prismatic element with triangular cross section.

Abstract: Whereas considerable effort has been expended in generating approximations to the spatial derivatives encountered in porous media flow, the time derivative has received relatively little attention. In spite of the fact that sophisticated finite element formulations have been developed for the spatial derivatives, finite difference methods are generally applied to the time derivative. The Galerkin method of approximation permits a high-order approximation in time as well as in space. The resulting approximate equations have been successfully solved by using a prismatic element with triangular cross section. The time axis runs the length of the prism and is subdivided into elements that may be linear, quadratic, or cubic. Because this formulation requires in general the solution for several time levels simultaneously, there is a resulting increase in computer time required to solve the larger matrix. Numerical experiments indicate that the selection of an optimum numerical scheme is dependent not only on the particular problem considered but also on the sequence of time steps used.