Further results on error estimators for local refinement with first-order system least squares (FOSLS)

Adaptive local refinement (ALR) can substantially improve the performance of simulations that involve numerical solution of partial differential equations. In fact, local refinement capabilities are one of the attributes of first-order system least squares (FOSLS) in that it provides an inexpensive but effective a posteriori local error bound that accurately identifies regions that require further refinement. Previous theory on FOSLS established the effectiveness of its local error estimator, but only under the assumption that the local region is not too 'thin'. This paper extends this theory to the case of a rectangular domain by showing that the estimator's effectiveness holds even for certain 'thin' local regions. Further, we prove that when the approximation satisfies a local saturation property, convergence of a FOSLS ALR scheme is guaranteed. Copyright (C) 2010 John Wiley & Sons, Ltd.