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This thesis deals with the hp-ﬁnite element method (FEM) for linear quadratic optimal control problems. Here, a tracking type functional with control costs as regularization shall be minimized subject to an elliptic partial diﬀerential equation. In the presence of control constraints, the ﬁrst order necessary conditions, which are typically used to ﬁnd optimal solutions numerically, can be formulated as a semi-smooth projection formula. Consequently, optimal solutions may be non-smooth as well. The hp-discretization technique considers this fact and approximates rough functions on ﬁne meshes while using higher order ﬁnite elements on domains where the solution is smooth.
The ﬁrst main achievement of this thesis is the successful application of hp-FEM to two related problem classes: Neumann boundary and interface control problems. They are solved with an a-priori reﬁnement strategy called boundary concentrated (bc) FEM and interface concentrated (ic) FEM, respectively. These strategies generate grids that are heavily reﬁned towards the boundary or interface. We construct an elementwise interpolant that allows to prove algebraic decay of the approximation error for both techniques. Additionally, a detailed analysis of global and local regularity of solutions, which is critical for the speed of convergence, is included. Since the bc- and ic-FEM retain small polynomial degrees for elements touching the boundary and interface, respectively, we are able to deduce novel error estimates in the L2- and L∞-norm. The latter allows an a-priori strategy for updating the regularization parameter in the objective functional to solve bang-bang problems.
Furthermore, we apply the traditional idea of the hp-FEM, i.e., grading the mesh geometrically towards vertices of the domain, for solving optimal control problems (vc-FEM). In doing so, we obtain exponential convergence with respect to the number of unknowns. This is proved with a regularity result in countably normed spaces for the variables of the coupled optimality system.
The second main achievement of this thesis is the development of a fully adaptive hp-interior point method that can solve problems with distributed or Neumann control. The underlying barrier problem yields a non-linear optimality system, which poses a numerical challenge: the numerically stable evaluation of integrals over possibly singular functions in higher order elements. We successfully overcome this diﬃculty by monitoring the control variable at the integration points and enforcing feasibility in an additional smoothing step. In this work, we prove convergence of an interior point method with smoothing step and derive a-posteriori error estimators. The adaptive mesh reﬁnement is based on the expansion of the solution in a Legendre series. The decay of the coeﬃcients serves as an indicator for smoothness that guides between h- and p-reﬁnement.

Ill-posed optimization problems appear in a wide range of mathematical applications, and their numerical solution requires the use of appropriate regularization techniques. In order to understand these techniques, a thorough analysis is inevitable.
The main subject of this book are quadratic optimal control problems subject to elliptic linear or semi-linear partial differential equations. Depending on the structure of the differential equation, different regularization techniques are employed, and their analysis leads to novel results such as rate of convergence estimates.