Adaptive Finite Element Computational Fluid Dynamics Using an Anisotropic Error Estimator

The aim of this paper is to present results on finite element adaptive strategies for computational fluid dynamics problems with singularities arising from shock phenomena and/or discontinuous boundary data. The adaptive analysis is based on an optimalmesh-adaptive strategy which is employed to refine the mesh, stretch and orient the elements in such a way that, along the adaptation process, the mesh becomes aligned with the discontinuities. This mesh adaptation process yields improved results in localizing regions of rapid or abrupt variations of the variables, whose location is not known a priori. On the other hand, the proposed mesh adaptation process is generated by minimizing, for a given number of elements in the mesh, a new anisotropic error estimator based on local directional interpolation error and recovering of the second derivatives of the finite element solution. Several adaptive mesh-refinement solutions for interpolation problems are also presented in order to show that the proposed optimal adaptive strategy using our anisotropic error estimator recovers optimal and/or superconvergent rates. Finally, applications of this approach to Computational Fluid Dynamics problems are also presented in order to show the computational performance of the proposed optimal adaptive procedure.