Method Galerkin

We have discussed already the application of the Galerkin method to discretization of integral operator equations (see Chapter 9). The same technique can be used for discretization of electromagnetic field differential equations as well. 

We introduce again a complex Hilbert space L2 (H) of the vector functions determined in the modeling region V and integrable in V with inner product  

In accord with standard practice, the region V in which calculations are to be made is divided into elements. For the two-dimensional case, the common practice is to divide the region into triangular elements, involving three points in the plane at a time. For three-dimensional problems, the equivalent element would be a tetrahedral element, D, involving four points in space. 

We introduce also a system of basis functions vi (r), V2 (r),. Vjv (r) in the Hilbert space L2 V). In the simplest case, we can use as basis functions the linear functions of coordinates, x, y, z, which are determined at any point within one of the elements, D, but are zero outside. We introduce a finite dimensional Hilbert subspace, L2 of the space L2 (D) C L2 D) spanned by the basis functions 

The second order electric or magnetic field equations (12.1) can be written in operator form as 

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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... 

Galerkin method becomes unstable and useless. It can also be seen that these oscillations become more intensified as a becomes larger (note that the factor affecting the stability is the magnitude of a and oscillatory solutions will also result using large negative coefficients). 

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 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... 

Donea, J., 1984.. A Taylor-Galerkin method for convective transport problems. Int. J. Nwn. Methods Eng. 20, 101 119. 

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). 

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. 

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). 

Petera, J., Nassehi, V. and Pittman,. T.F.T., 1993. Petrov-Galerkin methods on... 

Petera, J., Nassehi, V. and Pittman, J. F. T., 1993. Petrov Galerkin methods on isoparametric bilinear and biquadratic elements tested for a scalar convection-diffusion problem. Int. J. Numer. Methods Pleat Fluid Flow 3, 205-222. 

Let X G 2 be an arbitrary fixed element. Using the Galerkin method (see, for instance, Mikhailov, 1976), we can prove that there is a unique function 9 G Si satisfying the identity... 

So the necessary estimates are obtained, and we can use the Galerkin method to prove the solvability of the parabolic boundary value problem (5.185)-(5.188) (see Lions, 1969). This proves that the solution exists in the following sense. 

Element Matrices for Galerkin Method with Linear Shape Functions... 

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... 

When the coordinate functions y>iix) = y x — x )/h) are chosen by an approved rule as suggested before, the Ritz and the Bubnov-Galerkin methods coincide with the finite element method. 

Computational performance data on the Galerkin method were reported by Rachford and Dupont (Rl). Time increments of the order of 5 minutes are typical. According to these authors an average of 120 pipe steps per CPU second on a CDC 6600 computer was achieved. Approximately 100 words of storage per pipe were required for array storage and 12,000 words of storage for the program and subroutines. On that basis the authors estimated that a 1000-pipe network can be simulated on a full-size CDC 6600 computer at about 100 CPU seconds per simulated hour. 

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. 

This is the process that makes the method a Galerkin method. The basis for the orthogonality condition is that a function that is made orthogonal to each member of a complete set is then zero. The residual isT eing made orthogonal, and if the basis functions are complete and you use infinitely many of them, then the residual is zero. Once the residual is zero, tlie problem is solved. 

This equation defines the Galerkin method and a solution that satisfies this equation (for allj = 1,. ..,<= ) is called a weak solution. For an approximate solution, the equation is written once for each member of the trial function, j = 1,. .., NT — 1, and the boundary condition is applied. 

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. 

In the finite element method, Petrov-Galerkin methods are used to minimize the unphysical oscillations. The Petrov-Galerkin method essentially adds a small amount of diffusion in the flow direction to smooth the unphysical oscillations. The amount of diffusion is usually proportional to Ax so that it becomes negligible as the mesh size is reduced. The value of the Petrov-Galerkin method lies in being able to obtain a smooth solution when the mesh size is large, so that the computation is feasible. This is not so crucial in one-dimensional problems, but it is essential in two- and three-dimensional problems and purely hyperbolic problems. 

In the second approach, called the Galerkin method, one uses the property that the sampling functions satisfy the boundary conditions to write... 

The outline of this paper is as follows. First, a theoretical model of unsteady motions in a combustion chamber with feedback control is constructed. The formulation is based on a generalized wave equation which accommodates all influences of acoustic wave motions and combustion responses. Control actions are achieved by injecting secondary fuel into the chamber, with its instantaneous mass flow rate determined by a robust controller. Physically, the reaction of the injected fuel with the primary combustion flow produces a modulated distribution of external forcing to the oscillatory flowfield, and it can be modeled conveniently by an assembly of point actuators. After a procedure equivalent to the Galerkin method, the governing wave equation reduces to a system of ordinary differential equations with time-delayed inputs for the amplitude of each acoustic mode, serving as the basis for the controller design. 

The thin concentration boundary layer approximation, Eq. (3-51), has also been solved for bubbles k = 0) using surface velocities from the Galerkin method (B3) and from boundary layer theory (El5, W8). The Galerkin method agrees with the numerical calculations only over a small range of Re (L7). Boundary layer theory yields... 

The two most common of the methods of weighted residuals are the Galerkin method and collocation. In the Galerkin method, the weighting functions are chosen to be the trial functions, which must be selected as members of a complete set of functions. (A set of functions is complete if any function of a given class can be expanded in terms of the set.) Also according to Finlayson (1972),... 

Galerkin method forces the residual to be zero by making it orthogonal to each member of a... 

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