Introduction: Method of Weighted Residuals (MWR)
Credit!
This chapter is based on lecture notes originally prepared by Prof. Keith A. Woodbury of The University of Alabama. Further details can also be found in the book by [Grandin, 1991].
Suppose we have a linear differential operator
We wish to approximate
where
The notion in the MWR is to force the residual to zero in some average sense over the domain. That is
where the nuber of weight functions
Collocation method.
Sub-domain Method.
Least Squares method.
Galerkin method.
Each of these will be explained below. Afterwards we will apply some of them in an example to solve the steady-state advection diffusion equation.
Collocation Method
In this method, the weighting functions are taken from the family of Dirac
Hence the integration of the weighted residual statement results in the
forcing of the residual to zero at specific points in the domain. That is,
integration of equation (57) with
Sub-domain Method
This method doesn’t use weighting factors explicity, so it is not, strictly speaking, a member of the Weighted Residuals family. However, it can be considered a modification of the collocation method. The idea is to force the weighted residual to zero not just at fixed points in the domain, but over various subsections of the domain. To accomplish this, the weight functions are set to unity, and the integral over the entire domain is broken into a number of subdomains sufficient to evaluate all unknown parameters. That is
Least Squares Method
If the continuous summation of all the squared residuals is minimized, the rationale behind the name can be seen. In other words, a minimum of
In order to achieve a minimum of this scalar function, the derivatives of
Comparing with equation (57), the weight functions are seen to be
However, the
Galerkin Method
This method may be viewed as a modification of the Least Squares Method.
Rather than using the derivative of the residual with respect to the
unknown
Note that these are then identical to the original basis functions appearing in equation (55)