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We propose in this paper a multilevel correction method to solve optimalcontrol problems constrained by elliptic equations with the finite elementmethod. In this scheme, solving optimization problem on the finest finiteelement space is transformed to a series of solutions of linear boundary valueproblems by the multigrid method on multilevel meshes and a series of solutionsof optimization problems on the coarsest finite element space. Our proposedscheme, instead of solving a large scale optimization problem in the finestfinite element space, solves only a series of linear boundary value problemsand the optimization problems in a very low dimensional finite element space,and thus can improve the overall efficiency for the solution of optimal controlproblems governed by PDEs.

A multilevel correction method for optimal controls of elliptic equation