Asymptotically exact a posteriori error estimators for first-order div least-squares methods in local and global ^{L 2} norm

A new asymptotically exact a posteriori error estimator is developed for first-order div least-squares (LS) finite element methods. Let (^uh,σ_h) be LS approximate solution for (u,σ=-A ▿u). Then, E= ∥A-^1/2(σ_h+A ▿^uh) ∥0 is asymptotically exact a posteriori error estimator for ∥A1^/2 ▿(u-^uh) ∥0 or ∥A-^1/2(σ-σ_h) ∥0 depending on the order of approximate spaces for σ and u. For E to be asymptotically exact for ∥A1^/2 ▿(u-^uh) ∥0, we require higher order approximation property for σ, and vice versa. When both A ▿u and σ are approximated in the same order of accuracy, the estimator becomes an equivalent error estimator for both errors. The underlying mesh is only required to be shape regular, i.e., it does not require quasi-uniform mesh nor any special structure for the underlying meshes. Confirming numerical results are provided and the performance of the estimator is explored for other choice of spaces for (^uh,σ_h).