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

The contribution of this paper is twofold: firstly, a general approach to the goal-oriented a posteriori analysis of nonlinear partial differential equations is laid down, generalizing the standard DWR method to PetrovGalerkin formulations. This accounts 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; the effect of stabilization (e.g. for advectivedominated problems). 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 NavierStokes equations by continuous piecewise linear finite elements. The resulting adaptive procedure is thoroughly addressed and validated on some relevant test cases. This work has been supported by COFIN 2006 “Numerical Approximation of Multiscale and Multiphysics Problems with Adaptive Methods”.