Introduction to Finite Element Methods

The finite element methods are powerful techniques for the numerical solution of differential equations. Their theoretical formulation is based on the variational principle. The minimization of the functional of the form 

It has been shown that many differential equations that originate from the physical sciences have equivalent variational formulations. This is the basis for the well-known Rayleigh-Ritz procedure which in turn forms the basis for the finite element methods. 

An equivalent formulation of finite element methods can be developed using the concept of weighted residuals. In Sec. 5.6.3, we discussed the method of weighted residuals in connection with the solution of the two-point boundary-value problem. In that case we chose the solution of the ordinary differential equation as a polynomial basis function and caused the integral of weighted residuals to vanish  

We now extend this method to the solution of partial differential equations where the desired solution m(.) is replaced by a piecewise polynomial approximation of the form 

For a complete discussion of the variational formulation of the finite element method, see Vichnevetsky [3] and Vemuri and Karplus 6.  

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Donea, J. and Quartapelle, L., 1992. An introduction to finite element methods for transient advection problems. Comput. Methods Appl Meek Eng. 95, 169-203. 

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