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

TL;DR

This paper develops an adaptive edge element method for an electromagnetic constrained optimal control problem, providing convergence analysis and numerical validation of the method's effectiveness and quasi-optimality.

Contribution

It introduces a new adaptive algorithm with error estimators for an H(curl)-elliptic control problem and proves its convergence and quasi-optimality.

Findings

The adaptive method converges strongly to the exact solutions.

Error estimators effectively guide mesh refinement.

Numerical experiments confirm theoretical convergence and efficiency.

Abstract

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.