The finite element method

Both equations explained above are differential equations with respect to time and space. In order to solve these with numerical methods, we have to select a series of time instances and a finite set of variables to describe an approximate solution. In other words, both temporal and spatial discretisation is necessary. There is no difficulty in the one-dimensional time domain, it is sufficient to introduce an appropriate time step. However, the segmentation of the usually non-homogenous three-dimensional space is a problem of great concern. The finite difference, finite volume and finite element methods differ in the way the values of the selected computational variables are related to the approximate continuous solution. In case of the most general finite element method, the space is split up into small volumes, called elements. Within each element, the values are approximated as a linear combination of weighting or shape functions, where the coefficients are the computational variables [1]. If we substitute this approximate solution back into the flow and contaminant transport differential equations, the balance will not be identically equal to zero any more. What we get is the error function, a continuous function containing the computational variables. From this point, the task is to find those values for the variables that result in minimal error in some sense. Using the Galerkin method this can be done by solving a linear system of equations for every time step. However, the details exceed the scope of this paper.