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SUPERCONVERGENCE AND A POSTERIORI ERROR ESTIMATES OF VARIATIONAL DISCRETIZATION FOR ELLIPTIC CONTROL PROBLEMS
SUPERCONVERGENCE AND A POSTERIORI ERROR ESTIMATES OF VARIATIONAL DISCRETIZATION FOR ELLIPTIC CONTROL PROBLEMS
Journal of applied mathematics & informatics. 2014. Sep, 32(5_6): 707-719
  • Received : October 28, 2013
  • Published : September 30, 2014
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Hua Yuchun
Tang Yuelong

Abstract
In this paper, we investigate a variational discretization approximation of elliptic optimal control problems with control constraints. The state and the co-state are approximated by piecewise linear functions, while the control is not directly discretized. By using some proper intermediate variables, we derive a second-order convergence in <TEX>$L^2$</TEX>-norm and superconvergence between the numerical solution and elliptic projection of the exact solution in <TEX>$H^1$</TEX>-norm or the gradient of the exact solution and recovery gradient in <TEX>$L^2$</TEX>-norm. Then we construct a posteriori error estimates by using the superconvergence results and do some numerical experiments to confirm our theoretical results.
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