Connections between discontinuous Galerkin and nonconforming finite element methods for the Stokes equations

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Abstract

We study a discontinuous Galerkin finite element method (DGFEM) for the Stokes equations with a weak stabilization of the viscous term. We prove that, as the stabilization parameter γ tends to infinity, the solution converges at speed γ−1 to the solution of some stable and well-known nonconforming finite element methods (NCFEM) for the Stokes equations. In addition, we show that an a posteriori error estimator for the DGFEM-solution based on the reconstruction of a locally conservative H(div, Ω)-tensor tends at the same speed to a classical a posteriori error estimator for the NCFEM-solution. These results can be used to affirm the robustness of the DGFEM-method and also underline the close relationship between the two approaches. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011

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