A two‐level additive Schwarz method for the Morley nonconforming element approximation of a nonlinear biharmonic equation

In this paper, we consider the well known Morley nonconforming element approximation of a nonlinear biharmonic equation which is related to the well‐known two‐dimensional Navier–Stokes equations. Firstly, optimal energy and H¹‐norm estimates are obtained. Secondly, a two‐level additive Schwarz method is presented for the discrete nonlinear algebraic system. It is shown that if the Reynolds number is sufficiently small, the two‐level Schwarz method is optimal, i.e. the convergence rate of the Schwarz method is independent of the mesh size and the number of subdomains.