Finite Element Error Analysis for PDE-constrained Optimal Control Problems: The Control Constrained Case Under Reduced Regularity
Subject of this work is the analysis of numerical methods for the solution of optimal control problems governed by elliptic partial differential equations. Such problems arise, if one does not only want to simulate technical or physical processes but also wants to optimize them with the help of one or more influence variables. In many practical applications these influence variables, so called controls, cannot be chosen arbitrarily, but have to fulfill certain inequality constraints. The numerical treatment of such control constrained optimal control problems requires a discretization of the underlying infinite dimensional function spaces. To guarantee the quality of the numerical solution one has to estimate and to quantify the resulting approximation errors. In this thesis a priori error estimates for finite element discretizations are proved in case of corners or edges in the underlying domain and nonsmooth coefficients in the partial differential equation. These facts influence the regularity properties of the solution and require adapted meshes to get optimal convergence rates. Isotropic and anisotropic refinement strategies are given and error estimates in polygonal and prismatic domains are proved. The theoretical results are confirmed by numerical tests.
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adjoint velocity anisotropic approximation error Assumption FE3 Assumption PP1 Assumptions VAR1 bilinear form boundary value problem boundedness Cauchy-Schwarz inequality coeﬃcients comp conclude consider control problem 5.33 convergence rate Corollary Crouzeix-Raviart defined denote diﬀerent Dirichlet boundary conditions Dirichlet problem edge element error estimates embedding ﬁnite element error ﬁrst following lemma grading parameter holds independent of h interior angle interpolation operator L2-norm last step mesh grading mesh of type ndof Neumann boundary conditions Neumann problem norm optimal control problem Poisson equation polygonal domain postprocessing approach priori estimate prismatic domain Proof of Assumption proof of Lemma prove quasi-uniform meshes regularity results Rhu)u right-hand side second term singularities Sobolev embedding theorem Sobolev spaces Stokes equations tensor product Theorem 4.4 triangle inequality valid variational discrete approach weighted Sobolev spaces yields the assertion