Goal-Oriented Local A Posteriori Error Estimators for H(div) Least-Squares Finite Element Methods

Abstract

We propose a goal-oriented, local a posteriori error estimator for H(div) least-squares (LS) finite element methods. Our main interest is to develop an a posteriori error estimator for the flux approximation in a preassigned region of interest $D \subset \Omega$. The estimator is obtained from the LS functional by scaling residuals with proper weight coefficients. The weight coefficients are given in terms of local mesh size $h_T$ and a function $\omega_D$ depending on the distance to D. This new error estimator measures the pollution effect from the outside region of D and provides a basis for local refinement in order to efficiently approximate the solution in D. Numerical experiments show superior performances of our goal-oriented a posteriori estimators over the standard LS functional and global error estimators.