Anisotropic mesh adaptation in Computational Fluid Dynamics: application to the advection-diffusion-reaction and the Stokes problems

In this work we develop an anisotropic a posteriori error analysis of the advection–diffusion–reaction and the Stokes problems. This is the first step towards the study of more complex situations, such as the Oseen and Navier–Stokes equations, which are very common in Computational Fluid Dynamic (CFD) applications. The leading idea of our analysis consists of combining the anisotropic interpolation error estimates for affine triangular finite elements provided in [Numer. Math. 89 (2001) 641; Numer. Math. 94 (2003) 67] with the a posteriori error analysis based on a dual problem associated with the problem at hand [East–West J. Numer. Math. 4 (1996) 237; Comput. Methods Appl. Mech. Engrg. 166 (1998) 99]. Anisotropic interpolation estimates take into account in more detail the geometry of the triangular elements, i.e., not just their diameter but also their aspect ratio and orientation. On the other hand, the introduction of the dual problem allows us to control suitable functionals of the discretization error, e.g., the lift and drag around bodies in external flows, mean and local values, etc. The combined use of both approaches yields an adaptive algorithm which, via an iterative process, can be used for designing the optimal mesh for the problem at hand.