New a posteriori error estimate and quasi-optimal convergence of the adaptive nonconforming Wilson element

In this paper we establish the quasi-optimal convergence of the adaptive nonconforming Wilson element on the rectangular mesh. The main ingredients are a new a posteriori error estimator and a crucial observation that there is some special orthogonality between the conforming part and the nonconforming part in the energy inner product, which helps us to show the quasi-orthogonality and the discrete reliability. Finally we integrate these components in a usual way to achieve the quasi-optimal convergence.