Sharp error estimates for a finite element-penalty approach to a class of regulator problems

Authors:
Goong Chen, Wendell H. Mills, Shun Hua Sun and David A. Yost

Journal:
Math. Comp. **40** (1983), 151-173

MSC:
Primary 65K10; Secondary 49D30

DOI:
https://doi.org/10.1090/S0025-5718-1983-0679438-1

MathSciNet review:
679438

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Abstract: Quadratic cost optimal controls can be solved by penalizing the governing linear differential equation [2], [9]. In this paper, we study the numerical analysis of this approach using finite elements. We formulate the geometric *condition* (H) which requires that pairs of certain related finite-dimensional approximation spaces form “angles” which are bounded away from the “$180^\circ$ angle”. Under condition (H), we prove that the penalty parameter $\varepsilon$ and the discretization parameter *h* are independent in the error bounds, thereby giving sharp asymptotic error estimates. This condition (H) is shown to be also a necessary condition for such independence. Examples and numerical evidence are also provided.

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Article copyright:
© Copyright 1983
American Mathematical Society