HDG methods for the semilinear elliptic optimal control problem: a priori and a posteriori error estimates

Abstract

In this paper, we investigate a semilinear elliptic optimal control problem using a semidiscrete scheme in which the state and adjoint state variables are discretized by a hybridized discontinuous Galerkin (HDG) method, and the control variable is discretized by the variational discretization concept. For the HDG method applied to the semilinear elliptic equation, we prove a priori error estimates on quasi-uniform meshes and a posteriori error estimates on shape-regular meshes that satisfy the mesh condition (3.25). The global Lipschitz continuity on the nonlinear term is not required in the proofs. Then, these results are used to analyze the semilinear elliptic optimal control problem. As a consequence, by using the discrete functional analysis tools, we prove the existence of a subsequence of discrete solutions that converge to a solution of the continuous problem. Furthermore, for the control, state, and adjoint state variables, we prove a priori error estimates on quasi-uniform meshes and a posteriori error estimates on shape-regular meshes under the mesh condition (3.25) and the assumption that the second-order sufficient optimality conditions hold. Importantly, the proofs do not rely on the assumptions that the nonlinear term and its gradient are globally Lipschitz continuous, nor that the \(L^{\infty }\) norm of the difference between the adjoint state and its discretization is sufficiently small. Finally, we verify the theoretical results and show the robustness of the obtained a posteriori error estimators by two- and three-dimensional numerical examples.