Standard Galerkin method

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 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. 

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

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 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... 

Adopting the Galerkin method as a particular form of weighted residuals, i.e., considering the weights W, to be the same as the trial functions N after standard transformations of integrals in the relation (11), the next system of the ordinary differential equations with respect to nodal concentrations Q(t) may be derived ... 

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. 

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... 

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 ... 

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... 

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