Error estimates for the numerical approximation of Dirichlet boundary control for semilinear elliptic equations

We study the numerical approximation of boundary optimal control problems governed by semilinear elliptic partial differential equations with pointwise constraints on the control. The control is the trace of the state on the boundary of the domain, which is assumed to be a convex, polygonal, open set in R2. Piecewise linear finite elements are used to approximate the control as well as the state. We prove that the error estimates are of order O(h1−1/p) for some p > 2, which is consistent with the W1−1/p,p(Γ)-regularity of the optimal control

Convergencia de una familia de problemas discretos de control óptimo elíptico frontera respecto de un parámetro

Se considera un dominio acotado D de Rn con una frontera regular compuesta de dos porciones de frontera F1 y F2. En Gariboldi – Tarzia, Adv. Diff. Eq. Control Processes, 1 (2008), 113-132, se considera la convergencia de una familia de problemas de controles óptimos frontera de tipo Neumann gobernados por ecuaciones variacionales elípticas cuando el parámetro alpha de la familia (el coeficiente de transferencia de calor sobre la porción de frontera F1) tiende a infinito. Se demuestra la convergencia del control óptimo, del estado del sistema y del estado adjunto de la familia de problemas de controles óptimos fronteras de tipo Neumann a los correspondientes de un problema de control óptimo frontera de tipo Neumann también gobernado por una ecuación variacional elíptica con condiciones de contorno de tipo Dirichlet sobre F1. Se consideran, tanto para la familia de problemas de controles óptimos frontera de tipo Neumann como para el problema de control óptimo frontera límite, las aproximaciones numéricas por el método de los elementos finitos con triángulos de Lagrange de tipo 1. Se discretizan las ecuaciones variacionales elípticas que definen el estado del sistema y de su estado adjunto y además las funciones de costo respectivas. El objetivo del presente trabajo es el de estudiar la convergencia de la familia de problemas de controles óptimos fronteras de tipo Neumann discretos cuando el parámetro alpha tiende a infinito. Se demuestra la convergencia del control óptimo discreto, del estado del sistema discreto y del estado adjunto discreto de la familia a los correspondientes del problema de control óptimo frontera de tipo Neumann límite discreto.Fil: Tarzia, Domingo Alberto. Universidad Austral. Facultad de Ciencias Empresariales; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentin

A Frame Work for the Error Analysis of Discontinuous Finite Element Methods for Elliptic Optimal Control Problems and Applications to C0C^0 IP methods

In this article, an abstract framework for the error analysis of discontinuous Galerkin methods for control constrained optimal control problems is developed. The analysis establishes the best approximation result from a priori analysis point of view and delivers reliable and efficient a posteriori error estimators. The results are applicable to a variety of problems just under the minimal regularity possessed by the well-posed ness of the problem. Subsequently, applications of $C^0$ interior penalty methods for a boundary control problem as well as a distributed control problem governed by the biharmonic equation subject to simply supported boundary conditions are discussed through the abstract analysis. Numerical experiments illustrate the theoretical findings. Finally, we also discuss the variational discontinuous discretization method (without discretizing the control) and its corresponding error estimates.Comment: 23 pages, 5 figures, 1 tabl

A New HDG Method for Dirichlet Boundary Control of Convection Diffusion PDEs II: Low Regularity

In the first part of this work, we analyzed a Dirichlet boundary control problem for an elliptic convection diffusion PDE and proposed a new hybridizable discontinuous Galerkin (HDG) method to approximate the solution. For the case of a 2D polygonal domain, we also proved an optimal superlinear convergence rate for the control under certain assumptions on the domain and on the target state. In this work, we revisit the convergence analysis without these assumptions; in this case, the solution can have low regularity and we use a different analysis approach. We again prove an optimal convergence rate for the control, and present numerical results to illustrate the convergence theory