What are weighted residual approaches why are they used?

The method of weighted residuals (MWR) is an approximate technique for solving boundary value problems that utilizes trial functions satisfying the prescribed boundary conditions and an integral formulation to minimize error, in an average sense, over the problem domain.

What is the fundamental difference between residual and weighted residual methods?

Weighted residual methods are directly applied to the governing PDE of the system whereas in weak formulation is developed by the integration of weighted integral statement such that the order of derivative of the depend function is reduced.

Which is the best method for weighted residuals?

These five methods are: 1. collocation method. 2. Sub-domain method. 3. Least Squares method. 4. Galerkin method. 5. Method of moments. Each of these will be explained below. Two examples are then given illustrating their use. 2.1 Collocation Method In this method, the weighting functions are taken from the family of Dirac δfunctions in the domain.

What was the method of weighted residuals before finite element method?

Prior to development of the Finite Element Method, there existed an approximation technique for solving differential equations called the Method of Weighted Residuals (MWR).

What is the Galerkin method for weighted residuals?

In the method of weighted residuals, the next step is to determine appropriate weight functions. A common approach, known as the Galerkin method, is to set the weight functions equal to the functions used to approximate the solution. That is,

How to force the weighted residual to zero?

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