Standard Galerkin scheme

If the second and third terms in the weight function are neglected the standard Galerkin scheme will be obtained. 

The simplicity gained by choosing identical weight and shape functions has made the standard Galerkin method the most widely used technique in the finite element solution of differential equations. Because of the centrality of this technique in the development of practical schemes for polymer flow problems, the entire procedure of the Galerkin finite element solution of a field problem is further elucidated in the following worked example. 

The standard Galerkin technique provides a flexible and powerful method for the solution of problems in areas such as solid mechanics and heat conduction where the model equations arc of elliptic or parabolic type. It can also be used to develop robust schemes for the solution of the governing equations of... 

The standard least-squares approach provides an alternative to the Galerkin method in the development of finite element solution schemes for differential equations. However, it can also be shown to belong to the class of weighted residual techniques (Zienkiewicz and Morgan, 1983). In the least-squares finite element method the sum of the squares of the residuals, generated via the substitution of the unknown functions by finite element approximations, is formed and subsequently minimized to obtain the working equations of the scheme. The procedure can be illustrated by the following example, consider... 

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