Galerkin finite-elements method

Boundary value methods provide a description of the solution either by providing values at specific locations or by an expansion in a series of functions. Thus, the key issues are the method of representing the solution, the number of points or terms in the series, and how the approximation converges to the exact answer, i.e., how the error changes with the number of points or number of terms in the series. These issues are discussed for each of the methods finite difference, orthogonal collocation, and Galerkin finite element methods. 

Galerkin Finite Element Method In the finite element method, the domain is divided into elements and an expansion is made for the solution on each finite element. In the Galerkin finite element method an additional idea is introduced the Galerkin method is used to solve the equation. The Galerkin method is explained before the finite element basis set is introduced, using the equations for reaction and diffusion in a porous catalyst pellet. 

The Galerkin finite element method results when the Galerkin method is combined with a finite element trial function. The domain is divided into elements separated by nodes, as in the finite difference method. The solution is approximated by a linear (or sometimes quadratic) function of position within the element. These approximations are substituted into Eq. (3-80) to provide the Galerkin finite element equations. For example, with the grid shown in Fig. 3-48, a linear interpolation would be used between points x, and, vI+1. 

The set of partial differential equations developed for the simultaneous transfer of moisture, hear, and reactive chemicals under saturated/unsaturated soil conditions has been solved by the Galerkin finite element method. The chemical transport equations are formulated in terms of the total analytical concentration of each component species, and can be solved sequentially (Wu and Chieng, 1995). 

Parabolic Equations in One Dimension By combining the techniques applied to initial value problems and boundary value problems it is possible to easily solve parabolic equations in one dimension. The method is often called the method of lines. It is illustrated here using the finite difference method, but the Galerkin finite element method and the orthogonal collocation method can also be combined with initial value methods in similar ways. The analysis is done by example. 

FIG. 3-53 Trial functions for Galerkin finite element method linear polynomial on triangle. 

Peskin [49] used the Galerkin finite-element method to compute current distribution and shape change for electrodeposition into rectangular cavities. A concentration-dependent overpotential expression including both forward and reserve rate terms was used, and a stagnant diffusion layer was assumed. An adaptive finite-element meshing scheme was used to redefine the problem geometry after each time step. 

The Galerkin finite element method is successfully applied to flow in a relatively simple element of roll coating symmetric film-splitting in the nip region between smooth, rigid, counterrotating rolls. 

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... 

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 authors analyzed the problem by means of numerical analysis using the Galerkin finite element method, as well as a one-dimensional theory for viscoelastic filaments. Their findings were successfully used to interpret existing experimental data on Newtonian and viscoelastic jet... 

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