Advertisement

Concepts of Solution and the Finite Element Method: a Philosophical Take on Variational Crimes

  • Nicolas FillionEmail author
  • Robert M. Corless
Research Article

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.

Keywords

Mathematical solution Approximation Finite element method Variational crimes 

Notes

Acknowledgments

We would like to thank [removed for review]. We would also like to thank the two reviewers who made valuable suggestions.

References

  1. Ardourel, V., & Jebeile, J. (2017). On the presumed superiority of analytical solutions over numerical methods. European Journal for Philosophy of Science, 7(2), 201–220.CrossRefGoogle Scholar
  2. Bellen, A., & Zennaro, M. (2003). Continuous Runge–Kutta methods for ODEs. In Blah (Ed.) Numerical methods for delay differential equations: Oxford University Press.Google Scholar
  3. Bursten, J. (2015). Surfaces, scales and synthesis: scientific reasoning at the nanoscale. PhD thesis: University of Pittsburgh.Google Scholar
  4. Butcher, J.C. (2016). Numerical methods for ordinary differential equations. Wiley.Google Scholar
  5. Chen, D., Levin, D.I., Matusik, W., Kaufman, D.M. (2017). Dynamics-aware numerical coarsening for fabrication design. ACM Transactions on Graphics, 36(4), 84.CrossRefGoogle Scholar
  6. Corless, R.M. (1993). Six, lies, and calculators. The American Mathematical Monthly, 100(4), 344–350.CrossRefGoogle Scholar
  7. Corless, R.M., & Fillion, N. (2013). A graduate introduction to numerical methods, from the viewpoint of backward error analysis, (p. 868). New York: Springer.CrossRefGoogle Scholar
  8. Corless, R.M., & Fillion, N. (2019). Backward error analysis for perturbation methods. In Algorithms and complexity in mathematics, epistemology, and science (pp. 35–79): Springer.Google Scholar
  9. Courant, R. (1943). Variational methods for the solution of problems of equilibrium and vibrations. Bulletin of the American Mathematical Society, 43, 1–23.CrossRefGoogle Scholar
  10. Fillion, N., & Bangu, S. (2015). Numerical methods, complexity, and epistemic hierarchies. Philosophy of Science, 82, 941–955.CrossRefGoogle Scholar
  11. Fillion, N., & Corless, R.M. (2014). On the epistemological analysis of modeling and computational error in the mathematical sciences. Synthese, 191, 1451–1467.CrossRefGoogle Scholar
  12. Fillion, N., & Moir, R.H.C. (2018). Explanation and abstraction from a backward-error analytic perspective. European Journal for Philosophy of Science, 8 (3), 735–759.CrossRefGoogle Scholar
  13. Heyting, A. (1966). After thirty years. In Studies in logic and the foundations of mathematics, (Vol. 44 pp. 194–197): Elsevier.Google Scholar
  14. Higham, N.J. (2002). Accuracy and stability of numerical algorithms, 2nd edn. Philadelphia: SIAM.CrossRefGoogle Scholar
  15. Humphreys, P. (2004). Extending ourselves: computational science, empiricism and scientific method. USA: Oxford University Press.CrossRefGoogle Scholar
  16. Humphreys, P. (2009). The philosophical novelty of computer simulation methods. Synthese, 169(3), 615–626.CrossRefGoogle Scholar
  17. Ilie, S., Söderlind, G., Corless, R.M. (2008). Adaptivity and computational complexity in the numerical solution of ODEs. Journal of Complexity, 24(3), 341–361.CrossRefGoogle Scholar
  18. Kahan, W. (1980). Handheld calculator evaluates integrals. Hewlett-Packard Journal, 31(8), 23–32.Google Scholar
  19. Laymon, R. (1990). Computer simulations, idealizations and approximations. In PSA: Proceedings of the biennial meeting of the philosophy of science association, (Vol. 1990 pp. 519–534): Philosophy of Science Association.Google Scholar
  20. Moir, R.H. (2010). Reconsidering backward error analysis for ordinary differential equations. Master’s thesis: The University of Western Ontario.Google Scholar
  21. Morrison, M. (2009). Models, measurement and computer simulation: the changing face of experimentation. Philosophical Studies, 143, 33–57.CrossRefGoogle Scholar
  22. Peressini, A. (1997). Troubles with indispensability: applying pure mathematics in physical theory. Philosophia Mathematica, 5(3), 210–227.CrossRefGoogle Scholar
  23. Ramsey, J.L. (1992). Towards an expanded epistemology for approximations. In PSA: Proceedings of the biennial meeting of the philosophy of science association, (Vol. 1992 pp. 154–164): Philosophy of Science Association.Google Scholar
  24. Strang, G. (1972). Variational crimes in the finite element method. In Aziz, A. (Ed.) The mathematical foundations of the finite element method with applications to partial differential equations (pp. 689–710): Elsevier.Google Scholar
  25. Strang, G. (1973). Piecewise polynomials and the finite element method. Bulletin of the American Mathematical Society, 79(6), 1128–1137.CrossRefGoogle Scholar
  26. Strang, G., & Fix, G. (1973). Analysis of the finite element method. Prentice-Hall.Google Scholar
  27. Turing, A.M. (1952). The chemical basis of morphogenesis. Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences, 237(641), 37–72.CrossRefGoogle Scholar
  28. Werschulz, A.G. (1991). The computational complexity of differential and integral equations: an information-based approach. Oxford University Press.Google Scholar
  29. Wilkinson, J.H. (1971). Modern error analysis. SIAM Review, 13(4), 548–568.CrossRefGoogle Scholar
  30. Wilson, M. (1998). Mechanics, classical. In Craig, E. (Ed.) Routledge encyclopedia of philosophy. Routledge.Google Scholar
  31. Wilson, M. (2013). Enlarging one’s stall or how did all of these sets get in here? Philosophia Mathematica, 21, 2.CrossRefGoogle Scholar
  32. Wilson, M. (2017). Physics avoidance: and other essays in conceptual strategy. Oxford University Press.Google Scholar
  33. Winsberg, E. (2006). Handshaking your way to the top: simulation at the nanoscale. In Lenhard, J., Küppers, G., Shinn, T. (Eds.) Simulation: pragmatic constructions of reality (pp. 139–151): Springer.Google Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of PhilosophySimon Fraser UniversityBurnabyCanada
  2. 2.Department of Applied MathematicsWestern UniversityLondonUK

Personalised recommendations