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I am interested in the history of Finite Element Methods and Methods of Weighted Residuals (MWR), especially reduced quadrature and collocation methods. I have a paper coming out called “Orthogonal Collocation Revisited” which has a brief section on history of MWR and collocation methods. It will be available for a short time at https://authors.elsevier.com/a/1YHLy_12dr4lJw

I have found and read 7 articles on the history of FEM and a few presentations. One paper frequently cited as “a first” is Courant’s 1943 paper (based on 1941 presentation) “Variational Methods for the Solution of Problems of Equilibrium and Vibrations”. It seems the appendix of the paper is responsible for its citation as a first finite element paper. In the appendix he treats a torsion problem, first using a Raleigh-Ritz method with simple one and two term global trial functions. He then checks the results with a finite difference method on grids of triangles. He gives no details of the calculations. He also states:

“…. [the finite difference method] is obviously adaptable to any type of domain. Much more so than the Raleigh-Ritz procedure in which the construction of admissible functions would usually offer decisive obstacles.”

Since he does not use a variational method on grids of triangles and seems to think this would be difficult, why is the paper considered a first paper on the FEM?

This topic seems to have generated some interest, so I've added links to some of the information I've found online. I think the most even handed discussions are by Gupta and Meek and Oden.

Courant (1943) - http://mmph.narod.ru/doc/Courant.pdf

Strang (1973) - https://www.ams.org/journals/bull/1973-79-06/S0002-9904-1973-13351-8/S0002-9904-1973-13351-8.pdf

Williamson (1980) – https://ac.els-cdn.com/0315086080900014/1-s2.0-0315086080900014-main.pdf?_tid=60ec706d-33c0-4a27-b12e-9ac238c02390&acdnat=1549411660_5eb6ff6a29a71e8240751fc767691766

Oden (1987) - http://www.ce.memphis.edu/7117/notes/presentations/papers/Oden%20(1987)%20Historical%20Comments%20on%20Finite%20Elements.pdf

Gupta and Meek (1996) - http://people.sc.fsu.edu/~jpeterson/history_fem.pdf

Zienkiewicz (2004) – https://rodas5.us.es/file/3ca5a32a-9e22-ebb2-932e-1ce08a4ce607/1/birth_SCORM.zip/files/birth.pdf

Clough (2004) - https://onlinelibrary.wiley.com/doi/pdf/10.1002/nme.962

Samuelsson and Zienkiewicz (2006) - http://materiales.azc.uam.mx/gjl/Clases/EF10/2006_SamuelssonandZienkiewicz_History%20of%20the%20stiffness%20method.pdf

Gander and Wanner (2012) - https://www.unige.ch/~gander/Preprints/Ritz.pdf , Presentation - http://www-sop.inria.fr/nachos/seminars/2010/MGander-2010.pdf (this is one of several presentations by these authors, beware of German, French, Russian ….)

Finlayson and Scriven (1966) early history of Methods of Weighted Residuals - http://faculty.washington.edu/finlayso/MWR-AReview.pdf

L. Young
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  • I also posted this question at Stack Exchange - History of Science and Mathematics. Is there a way to link the two together? – L. Young Feb 06 '19 at 20:15

2 Answers2

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Because it is.

In Variational Methods for the Solution of Problems of Equilibrium and Vibrations, R. Courant gave an example of the variational formulation for the plate bending problem (Section I). After elaborating the connection with a class of minimization problem, he presented a method to approximate this problem numerically in Section II:

To construct a converging sequence of functions, each of which can be written as a finite linear combination of basis functions.

Essentially this is the soul of FEM, which is using a set of basis not nodal values, at least in my opinion.

He even remarked some of the phenomena on page 11 students would learn in a year-2000 finite element class: the convergence depends on the approximation space you choose (so-called consistency), but also depends on the problem itself (stability of the differential operator). He used the example of fourth order problem posing a much more challenging numerical task than the second order problem.

Moreover, technically speaking, it is not finite difference Courant checked in the appendix, as the "generalized finite difference" method does not use nodal values, instead it uses the expansion of the approximation function in a set of basis functions.

I am making this post a community wiki so that everyone can share his/her thoughts on this.

Shuhao Cao
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  • I don’t believe there is anything in Sections I and II that was not known before the 1940s. His discussion of boundary conditions is nice, but I believe that was all known even though there was (and still is) confusion about the proper imposition of boundary conditions. He does not mention anything about piecewise basis functions in Section II, which lies at the heart of the finite element method. The use of these methods with global basis functions was well established. – L. Young Feb 06 '19 at 19:31
  • Apparently, there is not universal agreement on what constitutes a finite element method. I make the distinction between global methods and finite element methods. A global method uses trial functions which are continuous across the entire domain, e.g. trig functions, whereas finite element methods are piecewise continuous, i.e. continuous within each element. Ritz, Galerkin and numerous others used global trial functions long before Courant's paper. I have noticed that many do not make this distinction. – L. Young Feb 12 '19 at 16:37
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I think I can answer my own question. I would still like to hear other comments on this topic. Since I was interested in FEM, I didn’t read the finite difference section III of the paper thoroughly the first time through.

Before 1943 there were many uses of variational or weighted residual methods to solve differential equations, but they used global trial or basis functions (usually trigonometric or polynomial). In my opinion, it became a FEM when these methods were combined with piecewise trial functions. As I understand it, the earliest engineering papers (e.g. Turner, Clough, Martin and Topp (1956)) built their approximations using heuristic arguments not variational principals. Apparently, Melosh (1963) was first to discover the approximations were equivalent to variational methods.

Many of the histories discuss the example in the appendix of Courant’s paper, which (as I state in my original post) discusses the variational method with global trial functions and finite differences on a triangular mesh. However, the key discussion in Courant’s paper is not the appendix, but on page 15 he states:

“If the variational problems contain derivatives not higher than the first order the method of finite difference can be subordinated to the Rayleigh-Ritz method by considering in the competition only [basis] functions [phi] which are linear in the meshes of a subdivision of our net into triangles formed by diagonals of the squares of the net.”

If he means first order in the weak form, then it would apply to second order differential equations and form the basis for the finite element method. If that is the case, why didn’t he use it in the Appendix? Why does he make several statements about the difficulty of constructing admissible basis functions?

The paper by Gupta and Meek takes the view that there was no single “first”, rather several papers contributed pieces to the method. Oden’s paper seems to take a similar view. I think it is fair to say that Courant’s paper was one of these pieces, but not the one and only “first”.

L. Young
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