An adaptive edge element method and its convergence for an electromagnetic constrained optimal control problem

In this work, an adaptive edge element method is developed for an H(curl)-elliptic constrained optimal control problem. We use the lowest-order Nedelec’s edge elements of first family and the piecewise (element-wise) constant functions to approximate the state and the control, respectively, and propose a new adaptive algorithm with error estimators involving both residual-type error estimators and lower-order data oscillations. By using a local regular decomposition for H(curl)-functions and the standard bubble function techniques, we derive the a posteriori error estimates for the proposed error estimators. Then we exploit the convergence properties of the orthogonal L^2- projections and the mesh-size functions to demonstrate that the sequences of the discrete states and controls generated by the adaptive algorithm converge strongly to the exact solutions of the state and control in the energy norm and L^2 -norm, respectively, by first achieving the strong convergence towards the solution to a limiting control problem. Three-dimensional numerical experiments are also presented to confirm our theoretical results and the quasi-optimality of the adaptive edge element method.

New Preconditioned Iteration Method Solving the Special Linear System from the PDE-Constrained Optimal Control Problem

In many fields of science and engineering, partial differential equation (PDE) constrained optimal control problems are widely used. We mainly solve the optimization problem constrained by the time-periodic eddy current equation in this paper. We propose the three-block splitting (TBS) iterative method and proved that it is unconditionally convergent. At the same time, the corresponding TBS preconditioner is derived from the TBS iteration method, and we studied the spectral properties of the preconditioned matrix. Finally, numerical examples in two-dimensions is applied to demonstrate the advantages of the TBS iterative method and TBS preconditioner with the Krylov subspace method.

Discretization of Optimal Control Problems Governed by p-Laplacian Elliptic Equations

In this paper, a state-constrained optimal control problem governed by p-Laplacian elliptic equations is studied. The feasible control set or the cost functional may be nonconvex, and the purpose is to obtain the convergence of a solution of the discretized control problem to an optimal control of the relaxed continuous problem.