An analogue of grad-div stabilization in nonconforming methods for incompressible flows

摘要

Grad-div stabilization is a classical remedy in conforming mixed finiteelement methods for incompressible flow problems, for mitigating velocityerrors that are sometimes called poor mass conservation. Such errors arise dueto the relaxation of the divergence constraint in classical mixed methods, andare excited whenever the spacial discretization has to deal with comparablylarge and complicated pressures. In this contribution, an analogue of grad-divstabilization is presented for nonconforming flow discretizations ofDiscontinuous Galerkin or nonconforming finite element type. Here the key isthe penalization of the jumps of the normal velocities over facets of thetriangulation, which controls the measure-valued part of the distributionaldivergence of the discrete velocity solution. Furthermore, we characterize thelimit for arbitrarily large penalization parameters, which shows that theproposed nonconforming Discontinuous Galerkin methods remain robust andaccurate in this limit. Several numerical examples illustrate the theory andshow their relevance for the simulation of practical, nontrivial flows.