Adaptive least-squares finite element methods: Guaranteed upper bounds and convergence in L2 norm of the dual variable

Abstract

We consider adaptive least-squares finite element methods. First, we develop a guaranteed upper bound for the dual error in the L2 norm, and this can be used as a stopping criterion for the adaptive procedures. Secondly, based on the a posteriori error estimates for the dual variable, we develop an error indicator that identifies the local area to refine, and establish the convergence of the adaptive procedures based on the Dörfler's marking strategy. Our convergence analysis is valid for the entire range of the bulk parameter 0<Θ≤1 and it shows the effect of bulk parameter and reduction factor of elements on the convergence rate. Confirming numerical experiments are provided.