2023/06/28 by Dongdong Liang, Wei Gong, Liang, Dongdong +3 · 1 citation
Computer Science · Engineering · Mathematics · #49J20 #65N15 #65N30 #Advanced Mathematical Modeling in Engineering #Advanced Numerical Methods in Computational Mathematics #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical Analysis (math.NA) #Optimization and Control (math.OC)
paper · pdf · doi:10.48550/arxiv.2306.15911
openalex publication_date 2023/06/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
In this paper, we consider the finite element approximation to a parabolic Dirichlet boundary control problem and establish new a priori error estimates. In the temporal semi-discretization we apply the DG(0) method for the state and the variational discretization for the control, and obtain the convergence rates O(k(1)/(4)) and O(k(3)/(4)-ε) (ε>0) for the control for problems posed on polytopes with y0∈ L2(Ω), yd∈ L2(I;L2(Ω)) and smooth domains with y0∈ H(1)/(2)(Ω), yd∈ L2(I;H1(Ω))∩ H(1)/(2)(I;L2(Ω)), respectively. In the fully discretization of the optimal control problem posed on polytopal domains, we apply the DG(0)-CG(1) method for the state and the variational discretization approach for the control, and derive the convergence order O(k(1)/(4) +h(1)/(2)), which improves the known results by removing the mesh size condition k=O(h2) between the space mesh size h and the time step k. As a byproduct, we obtain a priori error estimate O(h+k1\over 2) for the fully discretization of parabolic equations with inhomogeneous Dirichlet data posed on polytopes, which also improves the known error estimate by removing the above mesh size condition.