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ISSN 1088-6842(e) ISSN 0025-5718(p)

Superconvergence of mixed finite element methods for optimal control problems

Author: Yanping Chen
Journal: Math. Comp. 77 (2008), 1269-1291
MSC (2000): Primary 49J20, 65N30
Posted: February 28, 2008
MathSciNet review: 2398768
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Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we investigate the superconvergence property of the numerical solution of a quadratic convex optimal control problem by using rectangular mixed finite element methods. The state and co-state variables are approximated by the lowest order Raviart-Thomas mixed finite element spaces and the control variable is approximated by piecewise constant functions. Some realistic regularity assumptions are presented and applied to error estimation by using an operator interpolation technique. We derive superconvergence properties for the flux functions along the Gauss lines and for the scalar functions at the Gauss points via mixed projections. Moreover, global superconvergence results are obtained by virtue of an interpolation postprocessing technique. Thus, based on these superconvergence estimates, some asymptotic exactness a posteriori error estimators are presented for the mixed finite element methods. Finally, some numerical examples are given to demonstrate the practical side of the theoretical results about superconvergence.

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