Optimal control under reduced regularity

Optimal control of the Stokes equations: conforming and non-conforming finite element methods under reduced regularity

“…[10,19,23,24]. This paper extends the results of Rösch and Vexler [37] and Apel et al [3,6,7] to the optimal control of the Stokes equations under weaker regularity assumptions. This means, we do not assume that the velocity field is contained in W 1,∞ ( ) d ∩ H 2 ( ) d , but only in some weighted space H 2 ω ( ) (comp.…”

“…In the remainder of this section we give a couple of assumptions which are sufficient for proving a finite element error estimate for the optimal control problem (1)- (3). In order to discretize the optimal control problem, we consider a conforming triangulation T h of in the sense of Ciarlet [17], i.e.…”

“…They only depend on the structure of the underlying mesh as well as on the regularity of the solution of the optimal control problem (cf. (69), (A3) in [31], (4.4.2) in [39] or (37) in [3]). …”

“…Apel and co-authors got the same result for situations with reduced regularity in the state caused by corners and/or edges in the domain . They counteracted the singularities by isotropic [3,6] or anisotropic mesh grading [7]. In [8] the authors proved a convergence rate of h 2 ln h in L ∞ ( ) in plane domains for both, the post-processed as well as the variational-discrete control.…”

“…Therefore they restrict theirselves to polygonal, convex domains ⊂ R d , d = 2, 3, and assume in the case d = 3 that the edge openings of the domain are smaller than 2 3 π . The authors of [3,6,7] considered optimal control problems with scalar elliptic state equations also in non-convex domains, such that the state is not contained in W 1,∞ ( ) ∩ H 2 ( ). We do not only combine the techniques developed in these papers but also introduce substantially new things.…”