A new error analysis for parabolic Dirichlet boundary control problems
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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 y_0∈ L^2(Ω), y_d∈ L^2(I;L^2(Ω)) and smooth domains with y_0∈ H^1/2(Ω), y_d∈ L^2(I;H^1(Ω))∩ H^1/2(I;L^2(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(h^2) between the space mesh size h and the time step k. As a byproduct, we obtain a priori error estimate O(h+k^1 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.
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