Error estimate of a finite element method for an optimal control problem with corner singularity using the stress intensity factor

Abstract

We consider an optimal control problem for the Poisson equation on a non-convex polygonal domain with the corner singularity. Previously, we proposed a novel algorithm for the accurate numerical solution for the Poisson equation on a polygonal domain with the domain singularity. Then, we investigated the error estimate and its efficient procedure for the numerical algorithm. In this article, we propose an efficient algorithm and perform an error estimate for a distributed optimal control problem of the Poisson equation. The solutions of the optimality system with such singularity have singular decompositions: regular part plus singular part for each state variable and adjoint variable. The coefficient of the singular function is usually called stress intensity factor and can be computed by the extraction formula. We introduced a modified optimality system which has “zero” stress intensity factors using this stress intensity factor, from whose solutions we can compute very accurate solution of the original optimality system simply by adding a singular part. We give a precise error analysis and provide numerical results which justify the results therein.