A posteriori error estimates for the primary and dual variables for the div first-order least-squares finite element method

We propose a posteriori error estimators for first-order div least-squares (LS) finite element method for linear elasticity, Stokes equations and general second-order scalar elliptic problems. Our main interest is obtaining a posteriori error estimators for the dual variables (fluxes, strains, stress, etc.) which are main quantity of interest in many applications. We also provide a posteriori error estimators for the primary variable. These estimators are obtained from the local least-squares functional by assigning weight coefficients scaling the respective residuals. The weight coefficients are given in terms of local meshsize hK. We establish the global upper bounds and local lower bounds for the estimators. The estimators can be easily computed from the finite element solution together with the given problem data and provide basis for mesh refinement criteria for efficient computation of finite element solution (the indicators and estimators are identical). Numerical experiments show a superior performance of our a posteriori estimators for user-specific norm over the standard LS functional.