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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

Free boundaries and finite elements in one dimension


Authors: William W. Hager and Gilbert Strang
Journal: Math. Comp. 29 (1975), 1020-1031
MSC: Primary 65K05
MathSciNet review: 0388768
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Abstract: Two problems in control theory, one with state constraints and the other with control constraints, have been approximated by the finite element method. This discretization has been applied to both the primal and the dual formulation, in order to make a number of observations and comparisons:

1. The rate of convergence as the grid interval h is decreased, for polynomial elements of different degrees.

2. The presence or absence of a boundary layer in the error, concentrated at the "contact points" where the constraints change between binding and nonbinding.

3. The advantages of simpler constraints in the dual formulation, and the disadvantages of replacing strict convexity by ordinary convexity.

4. The numerical efficiency of each possible variation in achieving an approximate solution of reasonable accuracy.

We concluded that in our model problems, linear elements and the dual method provide the most efficient combination.


References [Enhancements On Off] (What's this?)

  • [1] William W. Hager, The Ritz-Trefftz method for state and control constrained optimal control problems, SIAM J. Numer. Anal. 12 (1975), no. 6, 854–867. MR 0415463 (54 #3550)
  • [2] W. W. HAGER, Rates of Convergence for Discrete Approximations to Problems in Control Theory, Ph. D. Thesis, M.I.T., June 1974.
  • [3] Gilbert Strang, The finite element method—linear and nonlinear applications, Proceedings of the International Congress of Mathematicians (Vancouver, B. C., 1974) Canad. Math. Congress, Montreal, Que., 1975, pp. 429–435. MR 0423842 (54 #11816)
  • [4] William W. Hager and Sanjoy K. Mitter, Lagrange duality theory for convex control problems, SIAM J. Control Optimization 14 (1976), no. 5, 843–856. MR 0410512 (53 #14261)
  • [5] William W. Hager, Rates of convergence for discrete approximations to unconstrained control problems, SIAM J. Numer. Anal. 13 (1976), no. 4, 449–472. MR 0500418 (58 #18058)

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Additional Information

DOI: http://dx.doi.org/10.1090/S0025-5718-1975-0388768-8
PII: S 0025-5718(1975)0388768-8
Article copyright: © Copyright 1975 American Mathematical Society