Overview

So far, we have learned how to solve partial differential equations using the Finite Differences Method (FDM). [Briggs et al., 2000] in their book “A Multigrid Tutorial” describe this as “…finite differences methods replace the problem domain by a grid and produce a vector whose components are approximations to the solution at the grid points”. This is straightforward and exactly what we have done in the past weeks. Here is what they have to say on finite elements: “…finite element methods partition the problem domain into subregions and produce a simple function in each subregion that approximates the solution”. This seems a bit less straightforward but results in a very powerful method to solve partial differential equations.

In this lecture, we will start out by having a closer look at finding approximate solutions for partial differential equations before diving deeper into finite element methods. A powerful technique, which is close to the finite element concept, is the Method of Weighted Residuals (MWR). This method will be presented as an introduction, before using a particular subclass of MWR, the Galerkin Method of Weighted Residuals, to derive the equations of the Finite Element Method (FEM).