Abstract
We consider elliptic distributed optimal control problems with energy
regularization. Here the standard
L
2
{L_{2}}
-norm regularization is
replaced by the
H
-
1
{H^{-1}}
-norm leading to more focused controls.
In this case, the optimality system can be reduced to a single singularly
perturbed diffusion-reaction equation known as differential filter in
turbulence theory. We investigate the error between
the finite element approximation
u
ϱ
h
{u_{\varrho h}}
to the state u and the
desired state
u
¯
{\overline{u}}
in terms of the mesh-size h and the
regularization parameter ϱ. The choice
ϱ
=
h
2
{\varrho=h^{2}}
ensures
optimal convergence the rate of which only depends
on the regularity of the target function
u
¯
{\overline{u}}
.
The resulting symmetric and positive definite system of finite element
equations is solved by the conjugate gradient (CG) method preconditioned
by algebraic multigrid (AMG) or balancing domain decomposition by
constraints (BDDC). We numerically study robustness and efficiency of the
AMG preconditioner with respect to h, ϱ, and the number of
subdomains (cores) p. Furthermore, we investigate the parallel
performance of the BDDC preconditioned CG solver.
Homogenization of constrained optimal control problems for one-dimensional elliptic equations on periodic graphs
