Dec 12, 2006, Tuesday, 03:40 pm, IAM, S-209
Ronald H. W. Hoppe
Dept. of Math., Univ. of Houston, USA; Inst. of Math., Univ. of Augsburg, Germany
We are concerned with an a posteriori error analysis of Adaptive Finite Element Methods (AFEMs) for distributed and boundary control problems with control constraints. The analysis is done for classes of 2D model problems using discretizations of the state and the co-state by conforming P1 elements and of the control and the co-control by element-wise constants with respect to shape-regular simplicial triangulations of the computational domain. The AFEMs consist of successive cycles of an adaptive loop involving the steps ’SOLVE’,’ESTIMATE’,’MARK’, and ’REFINE’. Here, ’SOLVE’ stands for the efficient solution of the discretized problems which is taken care of by appropriate primal-dual active set strategies. The following step ’ESTIMATE’ provides a residual-type a posteriori error estimator for the global discretization errors in the state, the co-state, the control, and the co-control. The error analysis further has to take into account oscillations in terms of the data of the problems. Based on a bulk criterion, in the step ’MARK’ we select edges and elements for refinement, whereas the final step ’REFINE’ is devoted to the technical realization of the refinement process.
The main results state convergence of the AFEMs and, under some assumption concerning the resolution of the free boundary, a guaranteed reduction of the discretization errors. Important ingredients of the convergence proofs are the reliability of the error estimator, its discrete local efficiency, and a perturbed Galerkin orthogonality. Numerical results illustrate the performance of the AFEMs.