Weighted residual finite element method

Weighted Residual Finite Element Methods - an Outline... 

WEIGHTED RESIDUAL FINITE ELEMENT METHODS - AN OUTLINE... 

Weighted residual finite element methods described in Chapter 2 provide effective solution schemes for incompressible flow problems. The main characteristics of these schemes and their application to polymer flow models are described in the present chapter. 

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... 

Derivation of the working equations of upwinded schemes for heat transport in a polymeric flow is similar to the previously described weighted residual Petrov-Galerkm finite element method. In this section a basic outline of this derivation is given using a steady-state heat balance equation as an example. 

Numerical methods such as the finite element method is based on the method of weighted residuals. To illustrate this method, we begin with boundary value partial differential equation (PDE) presented in the form... 

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... 

O. C. Zienkiewicz and C. Taylor, Weighted Residual Processes in Finite Elements with Particular Reference to Some Transient and Coupled Problems, in Lectures on Finite Element Methods in Continuum Mechanics, J. T. Oden and E. R. A. Oliveria, Eds., U. A. H. Press, Huntsville, AL, 1973. 

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... 

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... 

Finite element methods are based on global constraints imposed on the solution in certain, finite domains ( elements defining the discretization) along the space coordinate. Thus, the integrated residual between the true and the approximated solutions is forced to zero subject to a weighting function. 

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