Galerkin scheme

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... 

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). 

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

Petrov-Galerkin scheme - to discretize the energy Equation (5.25) for the calculation of T. 

O. Colin and M. Rudgyard. Development of high-order taylor-galerkin schemes for unsteady calculations. J. Comput. Phys., 162(2) 338-371, 2000. 

When reaction does take place and a > 0, the third boundary condition is nonlinear and to our knowledge no analytical solution to Eqs. (83 through (113 exists. To solve the equations, we used a finite difference Galerkin scheme (with the relaxation parameter set equal to 2/33 (29, 303. Numerical solutions were obtained for c as a function of x and y for different cases of Shyf, a and 0. The mixing cup concentration can be obtained from the solution of c and is given at any x by... 

The solution of the system (Equation 14.30) can be obtained numerically by time discretization. Because of the nonlinearities, the integrated mean value of the conduction matrix K = K T(t)) for a given period of time can be created and a numerical scheme of one or two time levels can be applied [96]. Favorable results have been obtained by the linear Galerkin scheme [97] ... 

To numerically solve Eq. 2, one can employ either the Galerkin scheme or the collocation scheme. In the Galerkin scheme, Eq. 2 is satisfied in a weighted integral sense, while in the collocation scheme, the integral equation is satisfied at the chosen collocation points. For simplicity, the collocation scheme is used to illustrate the numerical implementation of the BEM. 

Once the random field involved in the stochastic boundary value problem has been discretized, a solution method has to be adopted in order to solve the boundary value problem numerically. The choice of the solution method depends on the required statistical information of the solution. If only the first two statistical moments of the solution are of interest second moment analysis), the perturbafion method can be applied. However, if a. full probabilistic analysis is necessary, Galerkin schemes can be utilized or one has to resort to Monte Carlo simulations eventually in combination with a von Neumann series expansion. 

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... 

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 decoupled scheme the solution of the constitutive equation is obtained in a separate step from the flow equations. Therefore an iterative cycle is developed in which in each iterative loop the stress fields are computed after the velocity field. The viscous stress R (Equation (3.23)) is calculated by the variational recovery procedure described in Section 1.4. The elastic stress S is then computed using the working equation obtained by application of the Galerkin method to Equation (3.29). The elemental stiffness equation representing the described working equation is shown as Equation (3.32). 

In the consistent streamline upwind Petrov-Galerkin (SUPG) scheme all of the terms in Equation (3.52) are weighted using the function defined by Equation (3.53) and hence Wjj = Wj. 

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). 

Least-square.s and streamline upwind Petrov-Galerkin (SUPG) schemes... 

Retaining all of the terms in the w eight function a least-squares scheme corresponding to a second-order Petrov-Galerkin formulation will be obtained. 

In steady-state problems 6/S.l = 1 and the time-dependent term in the residual is eliminated. The steady-state scheme will hence be equivalent to the combination of Galerkin and least-squares methods. 

The described continuous penaltyf) time-stepping scheme may yield unstable results in some problems. Therefore we consider an alternative scheme which provides better numerical stability under a wide range of conditions. This scheme is based on the U-V-P method for the slightly compressible continuity equation, described in Chapter 3, Section 1.2, in conjunction with the Taylor-Galerkin time-stepping (see Chapter 2, Section 2.5). The governing equations used in this scheme are as follows... 

In generalized Newtonian fluids, before derivation of the final set of the working equations, the extra stress in the expanded equations should be replaced using the components of the rate of strain tensor (note that the viscosity should also be normalized as fj = rj/p). In contrast, in the modelling of viscoelastic fluids, stress components are found at a separate step through the solution of a constitutive equation. This allows the development of a robust Taylor Galerkin/ U-V-P scheme on the basis of the described procedure in which the stress components are all found at time level n. The final working equation of this scheme can be expressed as... 

Variational difference methods (the Ritz method and the Bubnov-Ga-lerkin method). The Ritz and the Bubnov-Galerkin variational methods have had considerable impact on complex numerical modeling problems and designs of difference schemes. 

So, the three-point scheme (30) (32) constructed by the Ritz method is identical with scheme (12) obtained by means of the IIM. In contrast to the Ritz method the Bubnov-Galerkin method applies equally well to... 

Equations (if.4) and (ff.S) are solved, along with the continuity equation (which does not change upon nondimensionalization), in a Cartesian coordinate system using the Fourier-Galerkin (spectral) technique under periodic boundary conditions in all three space dimensions. The scheme is similar to that used by Orszag [8] for direct solution of the incompressible Navier-Stokes equations. More details can be found in [9] and [7], and the scheme may be considered to be pseudospectral. ... 

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