CONVERGENCE OF DISCONTINUOUS GALERKIN APPROXIMATIONS OF AN OPTIMAL CONTROL PROBLEM ASSOCIATED TO SEMILINEAR PARABOLIC PDE'S

A discontinuous Galerkin finite element method for an optimal control problem related to semilinear parabolic PDE's examined. The schemes under consideration are discontinuous in time but confirming in space. Convergence of discrete schemes of arbitary order is proven. In addition, the convergence of discontinuous Galerkin approximates of the associated optimality system to the solutions of the continuous optimality system is shown. The proof is based on stability estimates at arbitrary time points under minimal regularity assumptions, and a discrete compactness argument for discontinuous Galerkin schemes (see Walkington [SINUM (June 2008) (submitted), preprint available at http://www.math.cmu.edu/similar to noelw], Sects. 3,4).