Standard Galerkin

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

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... 

Let us first consider the standard Galerkin solution of Equation (2.80) obtained using the previously described steps. 

The comparison between the finite element and analytical solutions for a relatively small value of a - 1 is shown in Figure 2.25. As can be seen the standard Galerkin method has yielded an accurate and stable solution for the differential Equation (2.80). The accuracy of this solution is expected to improve even further with mesh refinement. As Figmre 2.26 shows using a = 10 a stable result can still be obtained, however using the present mesh of 10 elements, for larger values of this coefficient the numerical solution produced by the standard... 

The first order derivative in Equation (2.80) corresponds to the convection in a field problem and the examples shown in Figure 2.26 illustraTes the ina bility of the standard Galerkin method to produce meaningful results for convection-dominated equations. As described in the previous section to resolve this difficulty, in the solution of hyperbolic (convection-dominated) equations, upwind-ing or Petrov-Galerkin methods are employed. To demonstrate the application of upwinding we consider the case where only the weight function applied to the first-order derivative in the weak variational statement of the problem, represented by Equation (2.82), is modified. 

The thermal conductivity of polymeric fluids is very low and hence the main heat transport mechanism in polymer processing flows is convection (i.e. corresponds to very high Peclet numbers the Peclet number is defined as pcUUk which represents the ratio of convective to conductive energy transport). As emphasized before, numerical simulation of convection-dominated transport phenomena by the standard Galerkin method in a fixed (i.e. Eulerian) framework gives unstable and oscillatory results and cannot be used. 

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

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

To illustrate the finite element method, the basic steps in the formulation of the standard Galerkin finite element method for solving a one-dimensional Poisson equation is outlined in the following. 

The finite element formulation is obtained using the Standard Galerkin method. 

The method in reference (4) to correct the pressure field for the effect of inertia was found to give reasonable convergence. In this case, the pressure is obtained by the standard Galerkin s method. The mean velocities U , Vm are calculated from non-inertial pressures. They do not satisfy the continuity equation, and therefore result in residual mass sources. These residuals together with the LHS of equation (4) produce the pressure correction p , which is then used to compute the velocity corrections u, v , as shown in Appendix 2. The process is repeated until the pressure fields do not change by 0.3%. In some cases, oscillations of the pressures behind the step occurred. This was smoothed partly by the upwind Galerkin s method, but it could not be completely removed. 

It was found that the standard Galerkin s method and the evaluation of the mass source term by the central difference technique gave stable results and they were adopted in this paper. 

Using the standard Galerkin finite element method, within any element the spatial distributions of the undeformed coordinates, velocities, and test functions are approximated using the same shape functions and nodal values as follows ... 

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,... 

---