WIAS Preprint No. 2448, (2017)

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


  • Akbas, Mine
  • Linke, Alexander
    ORCID: 0000-0002-0165-2698
  • Rebholz, Leo G.
  • Schroeder, Philipp W.

2010 Mathematics Subject Classification

  • 35Q30 65M15 65M60 76M10


  • Incompressible Navier--Stokes equations, mixed finite element methods, grad-div stabilization, Discontinuous Galerkin method, nonconforming finite elements




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

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