Output Functional Control for Nonlinear Equations Driven by Anisotropic Mesh Adaption: The Navier–Stokes Equations

Abstract

The contribution of this paper is twofold. Firstly, moving from the very well-known dual-weighted residual (DWR) method, we set up a theoretical framework for a goal-oriented a posteriori analysis of nonlinear partial differential equations accounting for different approximations of the primal and dual problems; nonhomogeneous Dirichlet boundary conditions, even different on passing from the primal to the dual problem; the error due to data approximation; and the effect of a possible stabilization. Secondly, moving from this framework and employing anisotropic interpolation error estimates, a sound anisotropic mesh adaption procedure is devised for the numerical approximation of the Navier–Stokes equations by continuous piecewise linear finite elements. The resulting adaptive procedure is thoroughly addressed and validated on some relevant test cases.