Goal-oriented adaptivity in pointwise state constrained optimal control of partial differential equations. (English) Zbl 1208.49025
Summary: We derive primal-dual weighted goal-oriented a posteriori error estimates for pointwise state constrained optimal control problems for second order elliptic partial differential equations. The constraints give rise to a primal-dual weighted error term representing the mismatch in the complementarity system due to discretization. In the case of sufficiently regular active (or coincidence) sets and problem data, a further decomposition of the multiplier into a regular \(L^2\)-part on the active set and a singular part concentrated on the boundary between the active and inactive set allows us to further characterize the mismatch error. The paper ends with a report on the behavior of the error estimates for test cases including the case of singular active sets consisting of only smooth curves or points.
MSC:
49K20 | Optimality conditions for problems involving partial differential equations |
65K10 | Numerical optimization and variational techniques |
65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
65N50 | Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs |
49M25 | Discrete approximations in optimal control |
90C46 | Optimality conditions and duality in mathematical programming |