A Perspective on Numerical Analysis of the Diffusion Equation

The equation of transient groundwater motion is founded on the principle of mass conservation and can be mathematically described by the diffusion equation. Recently, powerful integral formulations have been developed for numerically solving the diffusion equation under complex conditions. In the literature, it is customary to formulate the integral equations by integrating point differential equations. Instead. in this paper. we shall employ a direct method of formulation, starting from the concepts of set and measure, the notion of partitions and the definition of set-averages. When the direct approach is applied to formulate the well-known finite element (FEM) equations, it is seen that the Galerkin' weighting function, which is mathematically treated as an artifice for weighting residuals. is but an appropriate spatial partition function. The logical framework of the direct approach is then applied to study the properties of 'lumped' and consistent matrices arising in the use of the FEM. The lumped matrix, stemming naturally. from the direct approach, seeks to conserve mass locally as well as globally. while the consistent matrix, which results only when the differential equation is integrated in a specific. fashion. attempts only to preserve global mass balance. It is concluded that the direct approach is simple and complete and. in so far as the integral formulation is concerned, there is little to be gained in starting with the differential equation. Further. in formulating integral equations, it is common practice to evaluate only the time-dependent changes in the mass content of the system and ignore the evaluation of the mass content of the system at any given instant of time. In order to be complete in itself, a true integral approach should evaluate both the time-dependent changes in the mass content of the system as well as the instantaneous mass content at any given time.