An approximated solution of most problems described by a set of partial differential equations may be obtained by numerical methods like the finite element method (FEM), the discrete element method (DEM), the finite difference method (FDM), the finite volume method (FVM), or the boundary element method (BEM). For the problems concerned here, the most commonly used methods are the finite element and the finite difference procedures. Commonly, non-linear solid mechanics is better solved using the finite element method. Boundary element methods have strong limitation in the non-linear field. Finite difference methods are not easy to apply to tensorial equations (with the exception of the FLAC code, developed by Itasca).
Diffusion and advection-diffusion problems are often solved by finite difference or finite element methods. Some finite difference (or finite volume) codes are very popular for fluid flow, like e. g. MODFLOW, TOUGH2 (Pruess et al., 1999) for aquifer modelling or ECLIPSE (Schlumberger 2000) for oil reservoir modelling. These codes have been developed over a number of years and possess a number of specific features allowing users to take numerous effects into account. However, they suffer from some drawbacks, which limit their potential for modelling coupled phenomena. Therefore only a little information will be included here concerning finite difference approaches.
