Galerkin finite element schemes

It is commonly accepted that the finite element methods offer the most rigorous numerical schemes for the simulation of fluid flow phenomena. The inherent flexibility of these schemes and their ability to cope with complicated geometries and boundary conditions can be used very effectively to solve the governing equations of complex flow regimes. In particular, the finite element simulation of steady, incompressible laminar flow is very well-established, and an extensive literature in this area is available. Galerkin finite element schemes based on different types of Lagrange elements are the most frequently used techniques in these simulations [8]. In flow domains with porous walls, however, more recent work... 

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. 

Development of weighted residual finite element schemes that can yield stable solutions for hyperbolic partial differential equations has been the subject of a considerable amount of research. The most successful outcome of these attempts is the development of the streamline upwinding technique by Brooks and Hughes (1982). The basic concept in the streamline upwinding is to modify the weighting function in the Galerkin scheme as... 

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 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... 

Algorithms based on the last approach usually provide more flexible schemes than the other two methods and hence are briefly discussed in here. Hughes et al. (1986) and de Sampaio (1991) developed Petrov-Galerkin schemes based on equal order interpolations of field variables that used specially modified weight functions to generate stable finite element computations in incompressible flow. These schemes are shown to be the special cases of the method described in the following section developed by Zienkiewicz and Wu (1991). 

Extension of the streamline Petrov -Galerkin method to transient heat transport problems by a space-time least-squares procedure is reported by Nguen and Reynen (1984). The close relationship between SUPG and the least-squares finite element discretizations is discussed in Chapter 4. An analogous transient upwinding scheme, based on the previously described 0 time-stepping technique, can also be developed (Zienkiewicz and Taylor, 1994). 

One method to solve partial differential equations using the numerical schemes developed for solving time dependent ordinary differential methods is the method of lines. In this method, the spatial derivatives at time t are replaced by discrete approximations such as finite differences or finite element methods such as collocation or Galerkin. The reason for this approach is the advanced stage of development of schemes to solve ordinary differential equations. The resulting numerical schemes are frequently similar to those developed directly for partial differential equations. 

A. Onorati, M. Perotti, and S. Rebay. Modelling one-dimensional unsteady flows in ducts Symmetric finite difference schemes versus galerkin discontinuous finite element methods. International Journal of Mechanical Sciences, 39(11) 1213-1236, 1997. 

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