Among the numerical methods for solving governing equations representing the mathematical model of a problem, there is one very general and powerful method known as the “Finite Element Method (FEM)”. In this method, the discretization of the equations are carried out by dividing the solution domain into small sub-domains (elements), choosing a polynomial function to approximate the field variables over the element, and then minimizing the error resulting from this approximation.
In this chapter, we provide an introduction to FEM. Its general formulation is illustrated within the context of the general problems of flow of pore water and deformation of soil skeleton encountered in geotechnical engineering. The discussion starts with the Direct Stiffness Method (DSM), and is followed by a more general formulation of FEM presented using the Galerkin’s Method of Weighted Residual (GMWR). The necessary steps of obtaining a general finite element solution of a physical problem are also given. These are then applied to obtain numerical solutions of some geotechnical problems in both 1-D and 2-D. Among those, 1-D consolidation, 1-D coupled flow and deformation as well as 1-D and 2-D steady state seepage problems are the notable ones. MATLAB programs of all these problems and more are also provided.