Approximation of boundary control problems on curved domains

In this paper we consider boundary control problems associated to a semilinear elliptic equation defined in a curved domain Ω. The Dirichlet and Neumann cases are analyzed. To deal with the numerical analysis of these problems, the approximation of Ω by an appropriate domain Ωh (typically polygonal) is required. Here we do not consider the numerical approximation of the control problems. Instead, we formulate the corresponding infinite dimensional control problems in Ωh, and we study the influence of the replacement of Ω by Ωh on the solutions of the control problems. Our goal is to compare the optimal controls defined on Γ = ∂Ω with those defined on Γh = ∂Ωh and to derive some error estimates. The use of a convenient parametrization of the boundary is needed for such estimates

Analysis and Finite-Element Approximation of Optimal-Control Problems for the Stationary Navier-Stokes Equations with Distributed and Neumann Controls

We examine certain analytic and numerical aspects of optimal control problems for the stationary Navier-Stokes equations. The controls considered may be of either the distributed or Neumann type; the functionals minimized are either the viscous dissipation or the L4-distance of candidate flows to some desired flow. We show the existence of optimal solutions and justify the use of Lagrange multiplier techniques to derive a system of partial differential equations from which optimal solutions may be deduced. We study the regularity of solutions of this system. Then, we consider the approximation, by finite element methods, of solutions of the optimality system and derive optimal error estimates

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

Numerical analysis for the pure Neumann control problem using the gradient discretisation method

The article discusses the gradient discretisation method (GDM) for distributed optimal control problems governed by diffusion equation with pure Neumann boundary condition. Using the GDM framework enables to develop an analysis that directly applies to a wide range of numerical schemes, from conforming and non-conforming finite elements, to mixed finite elements, to finite volumes and mimetic finite differences methods. Optimal order error estimates for state, adjoint and control variables for low order schemes are derived under standard regularity assumptions. A novel projection relation between the optimal control and the adjoint variable allows the proof of a super-convergence result for post-processed control. Numerical experiments performed using a modified active set strategy algorithm for conforming, nonconforming and mimetic finite difference methods confirm the theoretical rates of convergence

On Asymptotic Global Error Estimation and Control of Finite Difference Solutions for Semilinear Parabolic Equations

The aim of this paper is to extend the global error estimation and control addressed in Lang and Verwer [SIAM J. Sci. Comput. 29, 2007] for initial value problems to finite difference solutions of semilinear parabolic partial differential equations. The approach presented there is combined with an estimation of the PDE spatial truncation error by Richardson extrapolation to estimate the overall error in the computed solution. Approximations of the error transport equations for spatial and temporal global errors are derived by using asymptotic estimates that neglect higher order error terms for sufficiently small step sizes in space and time. Asymptotic control in a discrete $L_2$-norm is achieved through tolerance proportionality and uniform or adaptive mesh refinement. Numerical examples are used to illustrate the reliability of the estimation and control strategies