Boundary Value Problems weighted residual methods

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... 

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. 

The method of weighted residuals has been used in solving a variety of boundary value problems, ranging from fluid flow to heat and mass transfer problems. It is popular because of the interactive nature of the first step, that is, the user provides a first guess at the solution and this is then forced to satisfy the governing equations along with the conditions imposed at the boundaries. The left-over terms, called residuah, arise because the chosen form of solution does not exactly satisfy either the equation or the boundary conditions. How these residual terms are minimized provides the basis for parameter or function... 

We have presented a family of approximate methods, called weighted residuals, which are quite effective in dealing with boundary value problems. The name suggests that we need to generate residuals obtained when the approximate solution is substituted into the governing equation. Then, we try to minimize the residuals or force it to be asymptotically close to zero at certain points. A number of methods have appeared, depending on how we minimize this residual. 

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. 

An equivalent formulation of finite element methods can be developed using the concept of weighted residuals. In Sec. 5.6.3, we discussed the method of weighted residuals in connection with the solution of the two-point boundary-value problem. In that case we chose the solution of the ordinary differential equation as a polynomial basis function and caused the integral of weighted residuals to vanish ... 

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