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$L^\infty$-error estimates of triangular mixed finite element methods for optimal control problems governed by semilinear elliptic equations

Zuliang Lu^a, Yanping Chen<sup>b

a</sup> Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Department of Mathematics, Xiangtan University, Xiangtan 411105, P.R. of China
^b School of Mathematical Sciences, South China Normal University, Guangzhou 510631, P.R. of China

Abstract: In this paper, we investigate $L^\infty$-error estimates for convex quadratic optimal control problems governed by nonlinear elliptic partial differential equations using mixed finite element methods. The state and the co-state are approximated by the lowest order Raviart-Thomas mixed finite element spaces and the control is approximated by piecewise constant functions. We derive $L^\infty$-error estimates of optimal order for a mixed finite element approximation of a semilinear elliptic optimal control problem. Finally, we present numerical tests which confirm our theoretical results.

Key words: $L^\infty$-error estimates, optimal control problem, semilinear elliptic equation, mixed finite element methods.

UDC: 517.93+519.713:007.52

Received: 02.06.2008
Revised: 17.07.2008

 English version: Numerical Analysis and Applications, 2009, 2:1, 74–86

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