Abstract
In this paper, we investigate the optimal control problems governed by elliptic equations with integral constraint for state variable in one dimension by Legendre---Galerkin spectral methods. We deduce optimal conditions of the optimal control problems. Meanwhile, we obtain an a priori error estimate and a posteriori error estimator. Furthermore, we obtain an explicit formula of the a posteriori error estimator by orthogonal properties of Legendre polynomials. Finally, we present numerical examples to confirm our analytical results.
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