Weighted residual statements in the context of finite element discretizations

1 Weighted residual statements in the context of finite element discretizations 

As already discussed, variations of a field unknown within a finite element is approximated by the shape functions. Therefore finite element discretization provides a nat ural method for the construction of piecewise approximations for the unknown functions in problems formulated in a global domain. This is readily ascertained considering the mathematical model represented by Equation (2.40). After the discretization of Q into a mesh of finite elements weighted residual statement of Equ tion (2.40), within the space of a finite element T3 , is written as 

For each IT), Equation (2.46) generates a corresponding equation and collectively these equations can be shown using matrix notation as 

In the standard Galerkin method (also called the Bubnov-Galerkin method) weight functions in the weighted residual statements are selected to be identical 

WEIGHTED RESIDUAL FINITE ELEMENT METHODS - AN OUTLINE 

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