A Posteriori Error Estimates for Semilinear Boundary Control Problems

In this paper we study the finite element approximation for boundary control problems governed by semilinear elliptic equations. Optimal control problems are very important model in science and engineering numerical simulation. They have various physical backgrounds in many practical applications. Finite element approximation of optimal control problems plays a very important role in the numerical methods for these problems. The approximation of optimal control by piecewise constant functions is well investigated by [7, 8]. The discretization for semilinear elliptic optimal control problems is discussed in [2]. Systematic introductions of the finite element method for optimal control problems can be found in [ 10]. As one of important kinds of optimal control problems, the boundary control problem is widely used in scientific and engineering computing. The literature on this problem is huge, see, e.g. [1, 9]. For some linear optimal boundary control problems, [11] investigates a posteriori error estimates and adaptive finite element methods. [ 3] discusses the numerical approximation of boundary optimal control problems governed by semilinear elliptic partial differential equations with pointwise constraints on the control. Although a priori error estimates and a posteriori error estimates of finite element approximation are widely used in numerical simulations, it is not yet been utilized in semilinear boundary control problems. Recently, in [4, 5, 6], we have derived a priori error estimates and superconvergence for linear optimal control problems using mixed finite element methods. A posteriori error analysis of mixed finite element methods for general convex optimal control problems has been addressed in [13]. In this paper, we derive a posteriori error estimates for a class of boundary control problems governed by semilinear elliptic equation. The problem that we are interested in is the following semilinear boundary control problems: