Abstract

In this paper, we study an elliptic optimal control problem with $$H^1$$H1-norm state constraint. The control problem is approximated by the Galerkin spectral method, which can provide high-order accuracy and fast convergence rate. The optimality conditions and a priori error estimates are presented. A reliable a posteriori error estimator is investigated, which is helpful for developing adaptive strategy in the spectral method. Some numerical tests confirm the error estimates and illustrate the performance of the indicator.

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