On the Adaptive Selection of the Parameter in Stabilized Finite Element Approximations

Abstract. A systematic approach is developed for the selection of the stabilization parameter for stabilized finite element approximation of the Stokes problem, whereby the parameter is chosen to minimize a computable upper bound for the error in the approximation. The approach is applied in the context of both a single fixed mesh and for an adaptive mesh refinement procedure. The optimization is carried out by a derivative free optimization algorithm (DFO) and is based on minimizing a new fully computable error estimator. Numerical results are presented illustrating the theory and the performance of the estimator together with the optimization algorithm.