Superconvergence for Neumann boundary control problems governed by semilinear elliptic equations

This paper is concerned with the discretization error analysis of semilinear Neumann boundary control problems in polygonal domains with pointwise inequality constraints on the control. The approximations of the control are piecewise constant functions. The state and adjoint state are discretized by piecewise linear finite elements. In a postprocessing step approximations of locally optimal controls of the continuous optimal control problem are constructed by the projection of the respective discrete adjoint state. Although the quality of the approximations is in general affected by corner singularities a convergence order of $h^2|\ln h|^{3/2}$ is proven for domains with interior angles smaller than $2\pi/3$ using quasi-uniform meshes. For larger interior angles mesh grading techniques are used to get the same order of convergence

Holographic entanglement entropy of local quenches in AdS4/CFT3: a finite-element approach

Understanding quantum entanglement in interacting higher-dimensional conformal field theories is a challenging task, as direct analytical calculations are often impossible to perform. With holographic entanglement entropy, calculations of entanglement entropy turn into a problem of finding extremal surfaces in a curved spacetime, which we tackle with a numerical finiteelement approach. In this paper, we compute the entanglement entropy between two half-spaces resulting from a local quench, triggered by a local operator insertion in a CFT3. We find that the growth of entanglement entropy at early time agrees with the prediction from the first law, as long as the conformal dimension Δ of the local operator is small. Within the limited time region that we can probe numerically, we observe deviations from the first law and a transition to sub-linear growth at later time. In particular, the time dependence at large Δ shows qualitative differences to the simple logarithmic time dependence familiar from the CFT2 case. We hope that our work will motivate further studies, both numerical and analytical, on entanglement entropy in higher dimensions

Error estimates for linear-quadratic control problems with control constraints

An abstract linear-quadratic optimal control problem is investigated with pointwise control constraints. This paper is concerned in discretization of the control by piecewise linear functions. Under the assumption that the optimal control and the optimal adjoint state is Lipschitz continuous and piecewise of class $C^2$ an approximation of order $h^{3/2}$ is proved for the solution of the control discretized problem with respect to the solution of the continuous one. Numerical tests are presented after the theoretical part

Error estimates for parabolic optimal control problems with control constraints

An optimal control problem for the 1-d heat equation is investigated with pointwise control constraints. This paper is concerned with the discretization of the control by piecewise linear functions. The connection between the solutions of the discretized problems and the continuous one is investigated. Under an additional assumption on the adjoint state an approximation order $\sigma^{3/2}$ is proved for uniform discretizations. In the general case it is shown that a non-uniform control discretization ensure an approximation of order $\sigma^{3/2}$. Numerical tests confirm the theoretical part

VII REUNIÓN ANUAL DE ASEPELT-ESPAÑA

VII Reunión Anual de ASEPELT-España, Cádiz, 17 y 18 de junio de 1993 - Cádi