Interior penalty variational multiscale method for the incompressible Navier–Stokes equation: Monitoring artificial dissipation

Abstract

In this paper, we apply the interior penalty method analyzed in Burman and Fernàndez [E. Burman, M. Fernàndez, Continuous interior penalty finite element method for the time-dependent Navier–Stokes equations: space discretization and convergence. Numer. Math. (2007)] to flows at high Reynolds number. As a possible measure of solution quality we propose to monitor the ratio between the artificial dissipation induced by the numerical method and the computed physical dissipation. For smooth flows we prove that for our method the artificial dissipation serves as an a posteriori error estimator and also that it vanishes at an optimal rate. In the case of flows with multiscale features, we discuss a heuristic approach relating the decay of the artificial dissipation to the decay rate of the power spectrum. Some numerical results in two space dimensions are presented examining the relation between the numerical dissipation and solution quality.