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We study [Formula presented] interior penalty methods for an elliptic optimal control problem with pointwise state constraints on two dimensional convex polygonal domains. The approximation of the optimal state is obtained by solving a fourth order variational inequality and the approximation of the optimal control is computed by a post-processing procedure. We prove the convergence of numerical solutions with rates in the [Formula presented]-like energy error by using the complementarity form of the variational inequality. Furthermore, we develop an a posteriori analysis for a residual based error estimator and introduce an adaptive algorithm. Numerical experiments are provided to gauge the performance of the proposed methods.

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Journal of Computational and Applied Mathematics

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