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**1 - 5**of**5**### ADAPTIVE DISCONTINUOUS GALERKIN METHODS FOR CONVECTION DOMINATED OPTIMAL CONTROL PROBLEMS

, 2012

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### Adaptive discontinuous Galerkin methods for state constrained optimal control problems governed by convection diffusion equations

, 2013

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### and M. Heinkenschloss: Local Error Analysis of Discontinuous Galerkin Methods for Advection-Dominated Elliptic Linear-Quadratic Optimal Control Problems [1].

"... repeated here. Theorem 5.1 of [1] is restated here as Corollary A.2. All references to equations and other results that do not start with ”A”, refer to the corresponding equations and results in [1]. We will analyze the error between the solution of the infinite dimensional optimal control problem ( ..."

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repeated here. Theorem 5.1 of [1] is restated here as Corollary A.2. All references to equations and other results that do not start with ”A”, refer to the corresponding equations and results in [1]. We will analyze the error between the solution of the infinite dimensional optimal control problem (2.5) and the solution of the discretized problem (3.6) in the presence of interior layers. The results in, e.g., [28, p. 473] or [29, L. 23.1] describe what parts of the forcing term f influence the exact solution of a single advection dominated PDE at any fixed point x0 ∈ Ω: The force term in the entire upstream direction of x0 influences the exact solution at x0, but only the force term from within an ε | log ε|-neighborhood in the streamline (downwind) direction and within a √ ε | log(ε)|-neighborhood in the crosswind direction influence exact solution at x0. The same behavior can be observed from the properties of the corresponding Green’s function. In the presence of interior layers only, the exact solution may vary strongly in the crosswind direction, but not in the streamline direction. Since the adjoint equation has similar properties, the same behavior of the solution can be expected from the coupled system. Our main goal of this section is to show that similarly to the case of a single equation (cf. [16]), the interior layers do not pollute the numerical solution to the coupled optimality system (3.7). We will accomplish this by weighted error estimates, where the purpose of the weighting function is essentially to isolate the domains of smoothness from the layers. The analysis is rather technical and in order to avoid unnecessary technicalities we will make several simplifications: • ε ≤ h, i.e. we consider only the advection-dominating case. • The reaction term r ≡ 1. This simplification is not essential, the same analysis can be applied to r(x) ≥ 0 (cf. Lemma 4.2).

### Adaptive Discontinuous Galerkin Approximation of Optimal Control Problems Governed by Transient Convection-Diffusion Equations

, 2015

"... In this paper, we investigate an a posteriori error estimate of a control constrained op-timal control problem governed by a time-dependent convection diffusion equation. Con-trol constraints are handled by using the primal-dual active set algorithm as a semi-smooth Newton method or by adding a More ..."

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In this paper, we investigate an a posteriori error estimate of a control constrained op-timal control problem governed by a time-dependent convection diffusion equation. Con-trol constraints are handled by using the primal-dual active set algorithm as a semi-smooth Newton method or by adding a Moreau-Yosida-type penalty function to the cost functional. An adaptive mesh refinement indicated by a posteriori error estimates is applied for both approaches. A symmetric interior penalty Galerkin method in space and a backward Euler in time are applied in order to discretize the optimization problem. Numerical results are presented, which illustrate the performance of the proposed error estimator. Impressum:

### Adaptive Symmetric Interior Penalty

, 2015

"... Galerkin method for boundary control ..."

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