The UNDERLYING mathematical basis of the finite ELEMENT method first lies with the classical Rayleigh-Ritz and variational calculus procedures. These theories provided the reasons why the finite element method worked well for the class of problems in which variational statements could be obtained (e.g., linear diffusion type problems). However,as interest EXPANDED in applying the finite element method to more types of problems, the use of classical theory to describe such problems became limited and could not be applied, e.g., fluid-related problems. Extension of the mathematical basis to non-linear and non-structural problems was achieved through the method of weighted residuals (MWR), originally conceived by GALERKIN in the early 20th century. The MWR was found to provide the ideal theoretical basis for a much wider basis of problems as opposed to the Rayleigh-Ritz method. Basically, the method requires the governing differential equation to be multiplied by a set of predetermined weights and the resulting product integrated over space; this integral is required to vanish. Technically, Galerkin's method is a subset of the general MWR procedure, since various types of weights can be utilized; in the case of Galerkin's method, the weights are chosen to be the same as the functions used to DEFINE the unknown variables. Most practitioners of the finite element method now employ Galerkin's method to establish the approximations to the governing equations.

The underlying mathematical basis of the finite element method first lies with the classical Rayleigh-Ritz and variational calculus procedures. These theories provided the reasons why the finite element method worked well for the class of problems in which variational statements could be obtained (e.g., linear diffusion type problems). However,as interest expanded in applying the finite element method to more types of problems, the use of classical theory to describe such problems became limited and could not be applied, e.g., fluid-related problems. Extension of the mathematical basis to non-linear and non-structural problems was achieved through the method of weighted residuals (MWR), originally conceived by Galerkin in the early 20th century. The MWR was found to provide the ideal theoretical basis for a much wider basis of problems as opposed to the Rayleigh-Ritz method. Basically, the method requires the governing differential equation to be multiplied by a set of predetermined weights and the resulting product integrated over space; this integral is required to vanish. Technically, Galerkin's method is a subset of the general MWR procedure, since various types of weights can be utilized; in the case of Galerkin's method, the weights are chosen to be the same as the functions used to define the unknown variables. Most practitioners of the finite element method now employ Galerkin's method to establish the approximations to the governing equations.