Abstract
In this paper, thea priorierror estimates of an embedded discontinuous Galerkin method for the Oseen equations are presented. It is proved that the velocity error in theL2(Ω) norm, has an optimal error bound with convergence orderk + 1, where the constants are dependent on the Reynolds number (orν−1), in the diffusion-dominated regime, and in the convection-dominated regime, it has a Reynolds-robust error bound with quasi-optimal convergence orderk + 1/2. Here,kis the polynomial order of the velocity space. In addition, we also prove an optimal error estimate for the pressure. Finally, we carry out some numerical experiments to corroborate our analytical results.
Highlights
As we know, numerous finite element methods are widely studied and applied for the incompressible NavierStokes equations
When H1-conforming finite elements for the velocity are used, the velocity errors in the L2 norm have been achieved with the quasi-optimal convergence order for some equalorder stabilized finite element methods, but the velocity errors aren’t pressure-robust, namely, the velocity error bounds are dependent on the pressure, see [3, 5, 7, 8, 12]
A simple modification is possible to use continuous facet function spaces for the trace velocity, which was wellknown as an embedded-hybridized discontinuous Galerkin (E-HDG) method [28]
Summary
Numerous finite element methods are widely studied and applied for the incompressible NavierStokes equations. An alternative to the modification of continuous finite element methods is H(div)-conforming discontinuous Galerkin (DG) method which was proved to have the quasi-optimal error estimate [1, 15], where the velocity spaces are H(div)-conforming. The HDG method introduced in [26] uses discontinuous facet function spaces for the trace velocity and pressure approximations. A simple modification is possible to use continuous facet function spaces for the trace velocity, which was wellknown as an embedded-hybridized discontinuous Galerkin (E-HDG) method [28]. The number of global coupled degrees of freedom can be reduced even further by using continuous facet function spaces for the trace velocity and pressure approximations, which was well-known as an embedded discontinuous Galerkin (EDG) method. The weak formulation is well posed by Babuska-Brezzi theory for all ν > 0 [2]
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