Weighted residual technique

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... 

Fitting model predictions to experimental observations can be performed in the Laplace, Fourier or time domains with optimal parameter choices often being made using weighted residuals techniques. James et al. [71] review and compare least squares, stochastic and hill-climbing methods for evaluating parameters and Froment and Bischoff [16] summarise some of the more common methods and warn that ordinary moments matching-techniques appear to be less reliable than alternative procedures. References 72 and 73 are studies of the errors associated with a selection of parameter extraction routines. 

Solution of a steady-state one-dimensional diffusion with a source term using Garlerkin and collocation weighted residual techniques. In this example, we want to solve the PDE... 

Lynn ei ai (1970) obtain the optimal temperature profiles for a tubular reactor with axial dispersion (Lynn et ai, 1970). Weighted residual techniques were used to solve the state and adjoint differential equations which result from the application of Pontryagin s maximum principle to the optimal control problem. 

Variational methods [5] are a class of high-order weighted residual techniques that combines the high spatial accuracy and rapid convergence of spectral methods with the generality and geometric flexibility of finite-element methods. Consider a variational method on Q for mie-dimensional Helmholtz Eq. 22. A variational formulation of this problem is that u(x) should be the solution to... 

Now applying the weighted residual technique to this equation gives ... 

The weighted residual method provides a flexible mathematical framework for the construction of a variety of numerical solution schemes for the differential equations arising in engineering problems. In particular, as is shown in the followmg section, its application in conjunction with the finite element discretizations yields powerful solution algorithms for field problems. To outline this technique we consider a steady-state boundary value problem represented by the following mathematical model... 

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... 

Following the procedure described in the continuous penalty technique subsection in Chapter 3 the Galerkin-weighted residual statements corresponding to Equation (4.52) are written as... 

Although the finite difference technique is generally easily implemented and is quite robust, the procedure often becomes numerically prohibitive for packed bed reactor models since a large number of grid points may be required to accurately define the solution. Thus, since the early 1970s most packed bed studies have used one of the methods of weighted residuals rather than finite differences. 

The method of weighted residuals comprises several basic techniques, all of which have proved to be quite powerful and have been shown by Finlayson (1972, 1980) to be accurate numerical techniques frequently superior to finite difference schemes for the solution of complex differential equation systems. In the method of weighted residuals, the unknown exact solutions are expanded in a series of specified trial functions that are chosen to satisfy the boundary conditions, with unknown coefficients that are chosen to give the best solution to the differential equations ... 

Of the various methods of weighted residuals, the collocation method and, in particular, the orthogonal collocation technique have proved to be quite effective in the solution of complex, nonlinear problems of the type typically encountered in chemical reactors. The basic procedure was used by Stewart and Villadsen (1969) for the prediction of multiple steady states in catalyst particles, by Ferguson and Finlayson (1970) for the study of the transient heat and mass transfer in a catalyst pellet, and by McGowin and Perlmutter (1971) for local stability analysis of a nonadiabatic tubular reactor with axial mixing. Finlayson (1971, 1972, 1974) showed the importance of the orthogonal collocation technique for packed bed reactors. 

The most common methodology when solving transient problems using the finite element method, is to perform the usual Garlerkin weighted residual formulation on the spatial derivatives, body forces and time derivative terms, and then using a finite difference scheme to approximate the time derivative. The development, techniques and limitations that we introduced in Chapter 8 will apply here. The time discretization, explicit and implicit methods, stability, numerical diffusion etc., have all been discussed in detail in that chapter. For a general partial differential equation, we can write... 

Other methods may be more appropriate for equations with particular mathematical characteristics or when more accurate, robust, stable and efficient solutions are required. The alternative spectral methods can be classified as sub-groups of the general approximation technique for solving differential equations named the method of weighted residuals (MWR) [51]. The relevant spectral methods are called the collocation Galerkin, Tan- and Least squares methods. These methods can also be applied to subdomains. The subdomain... 

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... 

The orthogonal collocation technique is a simple numerical method which is easy to program for a computer and which converges rapidly. Therefore it is useful for the solution of many types of second order boundary value problems. This method in its simplest form as presented in this section was developed by Villadsen and Stewart (1967) as a modification of the collocation methods. In collocation methods, trial function expansion coefficients are typically determined by variational principles or by weighted residual methods (Finlayson, 1972). The orthogonal collocation method has the advantage of ease of computation. This method is based on the choice of suitable trial series to represent the solution. The coefficients of trial series are determined by making the residual equation vanish at a set of points called collocation points , in the solution domain. 

In connection with the solutions of the equation of change, there will from time to time be found some new and useful analytical solutions. The emphasis surely will be put, however, on the development of more and more efficient numerical techniques, such as collocation and weighted residual methods, implemented with high-speed computers. 

In view of the solution of the potential problem - based on the method of weighted residual methods - a survey and discussion of the common solution techniques of the Laplace equation will be given. 

The method of weighted residuals (MWR) provides a general technique to choose the constants in such a... 

Equation (2.7) and the derived equations (2.8) and (2.9) are the weighted residual statements for all common approximating techniques of the Laplace equation. 

This means that the coefficients are found as the minimum of R, R). The least squares result is the most well-known criterion function for weighted residuals. The test function for this technique is more complicated, owing to the requirement of differentiation in Eq. 8.12. 

We illustrate the above five variations of weighted residuals with the following example of diffusion and first order chemical reaction in a slab catalyst (Fig. 8.1). We choose the first order reaction here to illustrate the five methods of weighted residual. In principle, these techniques can apply equally well to nonlinear problems, however, with the exception of the collocation method, the integration of the form (8.7) may need to be done numerically. 

Boundary value problems are encountered so frequently in modelling of engineering problems that they deserve special treatment because of their importance. To handle such problems, we have devoted this chapter exclusively to the methods of weighted residual, with special emphasis on orthogonal collocation. The one-point collocation method is often used as the first step to quickly assess the behavior of the system. Other methods can also be used to treat boundary value problems, such as the finite difference method. This technique is considered in Chapter 12, where we use this method to solve boundary value problems and partial differential equations. 

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