Standard Galerkin procedure

Standard Galerkin procedure - to discretize the circumferential component of the equation of motion, Equation (5.23), for the calculation of vs. 

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. 

This is Navier s equation of elastodynamics. Using the standard Galerkin method, one can obtain the weak form of this equation and then discretize the problem in space. This procedure entails the introduction of set of arbitrary functions 0, known as the test fimctions. The test functions are auxiliary fimctions which help formulate an approximate solution u to the displacements u, called the trial functions. The domain Q is then discretized in space using a set of global piecewise linear basis functions 4>, which divide the domain into discrete elements Q. As a result, both the test and trial functions become linear combinations of the global basis functions,... 

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... 

In the early 1970s, the standard finite element approximations were based upon the Galerkin formulation of the method of weighted residuals. This technique did emerge as a powerful numerical procedure for solving elliptic boundary value problems [102, 75, 53, 84, 50, 89, 17, 35]. The Galerkin finite element methods are preferable for solving Laplace-, Poisson- and and diffusion equations because they do not require that a variational principle exists for the problem to be analyzed. However, the power of the method is still best utilized in systems for which a variational principle exists, and it... 

In the upwind Galerkin method, the weighting function L/ = Nj + Fj, Fj is parabolic function of Ni, and they can be found in Appendix 1. Substitution of expression (9) into equation (8), and standard Hnite element procedure such as... 

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