Adaptive fully-discrete finite element methods for nonlinear quadratic parabolic boundary optimal control
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RESEARCH
Open Access
Adaptive fully-discrete finite element methods for nonlinear quadratic parabolic boundary optimal control Zuliang Lu* *
Correspondence: [email protected] School of Mathematics and Statistics, Chongqing Three Gorges University, Chongqing, 404000, P.R. China College of Civil Engineering and Mechanics, Xiangtan University, Xiangtan, 411105, P.R. China
Abstract The aim of this work is to study adaptive fully-discrete finite element methods for quadratic boundary optimal control problems governed by nonlinear parabolic equations. We derive a posteriori error estimates for the state and control approximation. Such estimates can be used to construct reliable adaptive finite element approximation for nonlinear quadratic parabolic boundary optimal control problems. Finally, we present a numerical example to show the theoretical results.
1 Introduction In this paper, we study the fully-discrete finite element approximation for quadratic boundary optimal control problems governed by nonlinear parabolic equations. Optimal control problems are very important models in 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 finite element approximation of a linear elliptic optimal control problem is well investigated by Falk [] and Geveci []. The discretization for semilinear elliptic optimal control problems is discussed by Arada, Casas, and Tröltzsch in []. Systematic introductions of the finite element method for optimal control problems can be found in [–]. As one of important kinds of optimal control problems, the boundary optimal control is widely used in scientific and engineering computing. The literature in this aspect is huge; see, e.g., [–]. For some quadratic boundary optimal control problems, Liu and Yan [, ] investigated a posteriori error estimates and adaptive finite element methods. Alt and Mackenroth [] were concerned with error estimates of finite element approximations to state constrained convex parabolic boundary optimal control problems. Arada et al. discussed the numerical approximation of boundary optimal control problems governed by semilinear elliptic equations with pointwise constraints on the control in []. Although a priori error estimates and a posteriori error estimates of finite element approximation are widely used in numerical simulations, they have not yet been utilized in nonlinear parabolic boundary optimal control problems. Adaptive finite element approximation is the most important method to boost accuracy of the finite element discretization. It ensures a higher density of nodes in a certain © 2013 Lu; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is prope
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