An improved numerical solution for the diffusion equation

This paper is concerned with the application of the two-dimensional Galerkin approximation to the modelling of the dynamic groundwater flow equation. Attention will be paid to a problem almost universally encountered in the modelling of these problems: the singularities caused by the presence of sources and/or sinks in the domain of interest. One method to improve the solution of the dynamic flow equation near boreholes, is to subtract the singular behaviour of flow with a time-dependent analytic solution for near-boreholes zones. By subtracting the singularities in this way, the resulting smooth problem may be solved by the Galerkin method and the result can be combined with the analytic solution. The result is an improved solution near boreholes without the need to use a refined mesh and thus a saving in computer storage and time.