Convergence of discontinuous Galerkin approximations of an optimal control problem associated to semilinear parabolic PDE's
National Technical University of
Athens, Department of Mathematics, Zografou Campus, Athens 15780,
Revised: 15 July 2009
A discontinuous Galerkin finite element method for an optimal control problem related to semilinear parabolic PDE's is examined. The schemes under consideration are discontinuous in time but conforming in space. Convergence of discrete schemes of arbitrary order is proven. In addition, the convergence of discontinuous Galerkin approximations 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/~noelw], Sects. 3, 4).
Mathematics Subject Classification: 65M60 / 49J20
Key words: Discontinuous Galerkin approximations / distributed controls / stability estimates / semi-linear parabolic PDE's.
© EDP Sciences, SMAI, 2009