Abstract
In this paper we present an extension of the continuous interior penalty method of Douglas and Dupont [Interior penalty procedures for elliptic and parabolic Galerkin methods, in Computing Methods in Applied Sciences, Lecture Notes in Phys. 58, Springer-Verlag, Berlin, 1976, pp. 207-216] to Oseen's equations. The method consists of a stabilized Galerkin formulation using equal order interpolation for pressure and velocity. To counter instabilities due to the pressure/velocity coupling, or due to a high local Reynolds number, we add a stabilization term giving L2 -control of the jump of the gradient over element faces (edges in two dimensions) to the standard Galerkin formulation. Boundary conditions are imposed in a weak sense using a consistent penalty formulation due to Nitsche. We prove energy-type a priori error estimates independent of the local Reynolds number and give some numerical examples recovering the theoretical results.
Highlights
The construction of finite element methods for the incompressible Navier–Stokes equations that are robust and accurate for a wide range of Reynolds numbers remains a challenging problem
In this paper we present an extension of the continuous interior penalty method of Douglas and Dupont [Interior penalty procedures for elliptic and parabolic Galerkin methods, in Computing Methods in Applied Sciences, Lecture Notes in Phys. 58, Springer-Verlag, Berlin, 1976, pp. 207–216] to Oseen’s equations
Note that by adding the L2-coercivity, we can use the stabilization term to control the convective term without using the H1-coercivity; this leads to a quasi-optimal estimate in the weaker L2-norm, with a ν-weighted H1 contribution showing that the stabilization handles the numerical instability induced by treating nonsymmetric terms using the standard Galerkin method
Summary
The construction of finite element methods for the incompressible Navier–Stokes equations that are robust and accurate for a wide range of Reynolds numbers remains a challenging problem. The convective term is controlled using a local inverse inequality, Lemma 4.4, Corollary 4.3, and the orthogonality of the L2-projection, after having replaced the continuous velocity field β by its piecewise linear interpolant βh,
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