The Finite Element Method as a Weighted Residual Method

One of the very popular numerical methods used for solving boundary value problems is the finite element method. A lot may be said and written about the method, but, here we are going to focus on the method as one that belongs to the weighted residual methods. To be more specific. The finite element method may be viewed as one of the Galerkin methods applied on a subdomain. The basic idea in this approach is that the differential equation may apply to any part, subdomain, of the whole domain, and we may apply the Galerkin method to that subdomain, called element. Further, the finite element method will use a function series that have generalized coordinates that have direct physical meaning, much like what we learner in the Lagrange interpolation method.

The first step in the finite element method is dividing the domain into elements (subdomains) that are connected to neighboring elements at nodes. The grid used by the finite element method is called unstructured grid because does not need to have any specific forder of numbering for the elements or the nodes (The Figure above presents an example of an unstructured grid creaded for aerodynamic analysis of an airfoil while the figure below presents a structured one). However, if we can impose some order to the numbering, that will result in better looking set of equations that may have some special methods of solving in a faster manner. In this work, we will not focus on ordering the elements and nodes in a certain manner, though, the problem we will introduce will be all having straight-forward numbering schemes.

For a one-dimensional domain. The grid generation is much simpler than what we see in the figures above. The domain is divided into intervals, elements, and at the end of each element the points are named nodes. The element length, thus, is the difference between the x-values of the end nodes. (see the figure below). Now that we have all what we need to know about the grid, we may start creating the interpolation functions.