Abstract
Despite being one of the most dependable methods used by applied mathematicians and engineers in handling complex systems, the finite element method commits variational crimes. This paper contextualizes the concept of variational crime within a broader account of mathematical practice by explaining the tradeoff between complexity and accuracy involved in the construction of numerical methods. We articulate two standards of accuracy used to determine whether inexact solutions are good enough and show that, despite violating the justificatory principles of one, the finite element method nevertheless succeeds in obtaining its legitimacy from the other.
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Notes
There are passing mentions of the finite element method in Peressini (1997), Winsberg (2006), Humphreys (2009), and Morrison (2009) and Bursten (2015), but no sustained discussion of the specific features of this method, or their philosophical implications. Without mentioning finite element methods explicitly, Wilson (e.g., 2013, 2017) has often discussed closely related themes.
Some would say that z is not a solution at all. This is incorrect if, as we do in this paper, one wishes to say that inexact solutions are solutions in some sense.
We added the labels (A)–(C) in the quote for future reference.
We remark that this proof relies on the discrete sampling necessary for computational work, and fails if the computer program is allowed to use interval arithmetic, or otherwise validate its results by faithfully interpolating between samples. This highlights the importance of interpolation that enables the computation of continuous numerical solutions. This also highlights the fact that, even though interval analysis is not a panacea to conceptually analyze the concept of approximation, approaches based on it (e.g., Laymon 1990) can capture this situation quite well.
If we use a continuously differentiable numerical method, such as the continuous explicit Runge-Kutta methods implemented in Matlab, we will then be in a position to compute the residual as well.
There are interesting additional issues. In a recent talk at UBC (where he presented results published in Chen 2017), D. M. Kaufman (Adobe Research and Columbia) pointed out that, in the context of computer graphics and animation, “realistic isn’t the same as accurate.” In some circumstances, such as solving systems with many millions of moving components, detailed accuracy may not be possible—but one can still get a realistic movie by presenting motions that “could have happened.” A good structured backward error can ensure this. There are many kinds of backward error, however, and they do need the proper physical context in order to make sense.
One often sees the maximum mesh diameter used; but with equidistribution of error, one recovers a mean mesh diameter. This does not seem widely known, and we will be pursuing this in future work. For the purposes of this paper, either the maximum mesh width or the mean mesh width, conditioned on equidistribution, can be considered to be going to zero in the asymptotic limit.
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Fillion, N., Corless, R.M. Concepts of Solution and the Finite Element Method: a Philosophical Take on Variational Crimes. Philos. Technol. 34, 129–148 (2021). https://doi.org/10.1007/s13347-019-00371-w
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DOI: https://doi.org/10.1007/s13347-019-00371-w