Error analysis for the finite element approximations of Dirichlet parabolic boundary control problem with measure data

Abstract

This article investigates the convergence analysis of finite element method for Dirichlet boundary control problem governed by parabolic equation with measure data. The presence of measure data in the source term and Dirichlet boundary control lead the solution of the state equation to have low regularity, which creates a challenging task for the error analysis. We prove the existence, uniqueness and regularity results of the solution to the control problem. For the discretization of the state and adjoint variables, we employ continuous piecewise linear polynomials while the control variable is approximated by piecewise constant functions. The backward Euler technique provides the foundation for the temporal discretization. For the completely discrete optimal control problem, a priori error bounds for the state variable in the L2(0,T;L2(Ω))-norm and for the control variable in the L2(0,T;L2(∂Ω))-norm are derived. Numerical experiment is performed to validate our theoretical analysis.