A POSTERIORI ERROR ESTIMATES OF hp-FEM FOR OPTIMAL CONTROL PROBLEMS

Finite element approximation plays an important role in the numerical methods of optimal control problems. There have been extensive theoretical and numerical studies for the finite element approximation of various optimal control problems. However the literature is too huge to even give a very brief review here. In recent years, the adaptive finite element method has been extensively investigated. Adaptive finite element approximation is among the most important means to boost the accuracy and efficiency of the finite element discretizations. It ensures a higher density of nodes in certain areas of the given domain, where the solution is more difficult to approximate, using an a posteriori error indicator. We acknowledge the pioneering work due to Babuška and Rheinboldt [4]. Further references can be found in the monographs [2], [5], [31], and the references cited therein. In the recent years, adaptive finite elements for optimal control has become a focus of research interests. There have appeared many research papers on the adaptivity of various optimal control problems. For example, [6] studied the adaptive finite element method for optimal control problems via a goal-orientated approach, while a posteriori error estimates of residual type were derived for convex distributed optimal control problems governed by the elliptic and the parabolic equations in [18], [22]-[24], and for boundary control problems in [21]. To authors’ knowledge, the papers discussing the adaptive finite element methods for optimal control problems are all related to low order FEM, i.e. h-FEM. In the adaptive h-FEM, the adaptivity is performed by mesh refinement guided by a posteriori error estimators. There are also many high order methods, such as spectral element methods, the p-version and the hp-version finite element methods, which have been applied to many practical problems. Using the local refinement of the meshes where the solution is singular and applying higher order polynomials where the solution is smooth, the adaptive hp-version finite element method can