Improved error estimates for semidiscrete finite element solutions of parabolic Dirichlet boundary control problems

Abstract

The parabolic Dirichlet boundary control problem and its finite element discretization are considered in convex polygonal and polyhedral domains. We improve the existing results on the regularity of the solutions by establishing and utilizing the maximal Lp-regularity of parabolic equations under inhomogeneous Dirichlet boundary conditions. Based on the proved regularity of the solutions, we prove O(h1−1/q0−ϵ) convergence for the semidiscrete finite element solutions for some q0>2⁠, with q0 depending on the maximal interior angle at the corners and edges of the domain and ϵ being a positive number that can be arbitrarily small.