Variational Multiscale error estimator for anisotropic adaptive fluid mechanic simulations: Application to convection-diffusion problems

In this work, we present a new a posteriori error estimator based on the Variational Multiscale method for anisotropic adaptive fluid mechanics problems. The general idea is to combine the large scale error based on the solved part of the solution with the sub-mesh scale error based on the unresolved part of the solution. We compute the latter with two different methods: one using the stabilizing parameters and the other using bubble functions. We propose two different metric tensors Hiso and Hanisonew. They are both defined by the recovered Hessian matrix of the solution and rely on the sub-grid scale error estimator. Thus, we develop a new anisotropic local error indicator and we test it for mesh adaptation on convection-dominated benchmarks in 2D and 3D. The results show that the proposed error indicator leads to enhanced and accurate solutions while using a drastically reduced number of elements.