Analysis of the Discontinuous Galerkin Interior Penalty Method with Solenoidal Approximations for the Stokes Equations

Discontinuous Galerkin (DG) methods have become very popular for incompressible flow problems, especially in combination with piecewise solenoidal approximations [2, 4, 5, 7, 12, 13, 14, 15]. In the context of conforming finite elements, solenoidal approximations were derived by Crouzeix and Raviart in [6], allowing one to obtain a formulation involving only velocity. Nevertheless their implementation is non-trivial and they are limited to low-order approximations. While alternative solutions for incompressible flows are, among others, velocity-pressure formulations satisfying the Babuska-Brezzi condition, or hp-version FEM, in a DG framework high-order solenoidal approximations can be easily defined. This leads to an important saving in the number of degrees of freedom, with the corresponding reduction in computational cost, see [16]. Cockburn and collaborators [4, 5] were among the first researchers to use solenoidal approximations for incompressible flows in the context of the Local Discontinuous Galerkin (LDG) method, and they also introduced the concept of hybrid pressure. Later, the use of solenoidal approximations and hybrid pressure has been applied to an Interior Penalty Method (IPM), in [13], and to a Compact Discontinuous Galerkin (CDG) method, see [16, 17]. In [13], the velocity approximation space is decomposed in every element into a solenoidal part and an irrotational part. This allows for a splitting of the original weak form in two uncoupled problems. The first one solves for velocity and hybrid pressure, and the second one allows evaluating the pressure in the interior of the elements, as a post-processing of the velocity solution. LDG, CDG and IPM methods all lead to symmetric and coercive bilinear forms for self-adjoint operators. But IPM and CDG methods have the major advantage, relative to LDG, of being compact formulations, that is, the degrees of freedom of one element are only connected to those of immediate neighbors. In [16], IPM and CDG methods are further compared for the solution of the Navier-Stokes equations.