An adaptive finite element method for distributed elliptic optimal control problems with variable energy regularization
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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 ϱ is replaced by a suitable representation of the energy norm in H^-1(Ω) involving a variable, mesh-dependent regularization parameter ϱ(x). It turns out that the error between the computed finite element state u_ϱ h and the desired state u̅ (target) is optimal in the L^2(Ω) norm provided that ϱ(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_ϱ h - u̅_L^2(Ω) between the finite element state u_ϱ h and the target u̅. 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.
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