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Recently, I took a course on numerical methods where we learned about Galerkin's Method of Weighted Residuals. We were told that it forms the basis of the Finite Element Method .

However, in most introductory texts I've seen on FEM, they start with a simple example where they manually write out the equations to minimize the potential energy at each node. For instance, see this example of a system of springs. There are no test functions, so it doesn't seem to be the MWR.

So, my question is:

  1. Is there a name for this other method?
  2. What are the practical and numerical differences between it and the Galerkin Method of Weighted Residuals? (e.g., is it easier to set up for certain problems? Does it have better numerical precision or stability?)
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  • $\begingroup$ The spring (or in higher dimensions, truss) examples are presented first since they are easy to think about, especially for the undergraduate mechanical engineers for whom many of these introductory texts are written. They serve to show the basic workflow of a finite element solution (element tangent --> assembly --> solution) without muddying the water with test functions and other formalities, which can be addressed in detail later. Once you're comfortable with the actual mathematics involved, you can probably safely ignore the physical example system. $\endgroup$
    – Tyler Olsen
    Mar 20, 2016 at 17:43
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    $\begingroup$ The finite element method is a specific case of Galerkin's method, defined by a special choice of piecewise polynomial functions as test and trial functions. I already gave my opinion on introducing finite element methods via systems of springs or similar in this answer (contrary to @TylerOlsen, I think this muddies the water rather than the other way around) -- the test functions are there as well, but hidden in the way the systems are assembled in a purely local fashion. $\endgroup$
    – Christian Clason
    Mar 20, 2016 at 19:19
  • $\begingroup$ @ChristianClason Just read the other answer, and that's fair enough, but you'd have to be in a math class for that approach. I had it introduced to me in a structural mechanics class, so the audience was right for the mechanical interpretation, I suppose. $\endgroup$
    – Tyler Olsen
    Mar 20, 2016 at 19:34
  • $\begingroup$ 1. I often hear "this other method" referred to as the direct stiffness method. 2. Weighted residuals exploits the strong form of the PDE, while FEM exploits the weak form of the PDE. See this answer for more details. $\endgroup$
    – Paul
    Mar 20, 2016 at 20:32
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    $\begingroup$ I think your question already has an answer here: scicomp.stackexchange.com/questions/7845/… $\endgroup$
    – Paul
    Mar 22, 2016 at 1:30

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