An adaptive finite element method for distributed elliptic optimal control problems with variable energy regularization

Abstract

We analyze the finite element discretization of distributed elliptic optimal control problems with variable energy regularization, where the usual L 2 (Ω) norm regularization term with a constant regularization parameter is replaced by a suitable representation of the energy norm in H −1 (Ω) involving a variable, mesh-dependent regularization parameter (x). It turns out that the error between the computed finite element state u h and the desired state u (target) is optimal in the L 2 (Ω) norm provided that (x) behaves like the local mesh size squared. This is especially important when adaptive meshes are used in order to approximate discontinuous target functions. The adaptive scheme can be driven by the computable and localizable error norm u h − u L 2 (Ω) between the finite element state u h and the target u. The numerical results not only illustrate our theoretical findings, but also show that the iterative solvers for the discretized reduced optimality system are very efficient and robust.

Robust Finite Element Discretization and Solvers for Distributed Elliptic Optimal Control Problems

“…The comparison of convergence on both uniform and adaptive refinements is given in Figure 1. We observe the convergence rate h 0.5 for the uniform refinement as predicted by Theo-rem3, and a much better convergence rate h 0.75 for the adaptive refinements; see [21] for the case of variable energy regularization. There one can also find an explanation of the convergence rate that can be achieved via this adaptive procedure.…”

“…Further, we run tests on the adaptively refined meshes, in which we have employed the standard red-green refinement of tetrahedral elements, and we have chosen the locally varying regularization parameter τ = h 4 e on each tetrahedral element τ e . The adaptive procedure is simply based on the localization of the error y d −ỹ L2(Ω) that is explicitly computable for any known fe approximation ỹ to the given desired state y d ; see [21] for a detailed description.…”