Hp Discontinuous Galerkin Approximations for the Stokes Problem

Abstract

We propose and analyze a discontinuous Galerkin approximation for the Stokes problem. The finite element triangulation employed is not required to be conforming and we use discontinuous pressures and velocities. No additional unknown fields need to be introduced, but only suitable bilinear forms defined on the interfaces between the elements, involving the jumps of the velocity and the average of the pressure. We consider hp approximations using Q k -Q k velocity-pressure pairs with k = k + 2, k + 1, k. Our methods show better stability properties than the corresponding conforming ones. We prove that our first two choices of velocity spaces ensure uniform divergence stability with the respect to the mesh size h. Numerical results show that they are uniformly stable with respect to the local polynomial degree k, a property that has no analog in the conforming case. An explicit bound in k which is not sharp is also proven. Numerical results show that if equal order approximation is chosen for the velocity and pressure, no spurious pressure modes are present but the method is not uniformly stable either with respect to h or k. We derive a priori error estimates generalizing the abstract theory of mixed methods. Optimal error estimates in h are proven. As for discontinuous Galerkin methods for scalar diffusive problems, half power of k is lost for p and hp approximations independently of the divergence stability.Keywords: Mixed problems, hp approximations, spectral elements, discontinuous Galerkin approximations, non-conforming approximations Subject Classification: 65N30, 65N35, 65N12, 65N15 1. Introduction. Discontinuous Galerkin (DG) methods have a long history and have recently become more and more popular. They have been heavily tested and studied, and they present considerable advantages for certain types of problems, especially those modeling phenomena where convection is strong; see the monograph [15].Their main idea relies in the choice of approximation spaces consisting of piecewise polynomial functions with no kind of continuity constraints across the interface between the elements of a triangulation. Consistency and well-posedness are achieved by introducing suitable bilinear forms defined on the interface. In this respect they are closely related to finite volume methods as they relies on the definition of numerical fluxes. As for conforming finite element approximations, the corresponding discrete problem is given in terms of finite dimensional subspaces and bilinear forms.One of the main advantages of DG methods is that they allow a much greater flexibility in the design of the mesh and in the choice of the approximation spaces. Indeed, if one abandons the idea of a conforming approximation, one may as well abandon the idea of a conforming triangulation. This was soon realized and exploited in some DG methods; see, e.g., [19]. A mixed domain decomposition approach is also natural where conforming approximations are considered on single subdomains or patches, and DG interface terms are introduc...

Discontinuous Galerkin methods for the Stokes equations using divergence‐free approximations

“…This weak form is close to the formulation proposed in [7] where stability is also studied. It clearly identifies pressure with the Lagrange multiplier that imposes both a weakly solenoidal field inside each element and a continuous normal component along Γ.…”

“…It clearly identifies pressure with the Lagrange multiplier that imposes both a weakly solenoidal field inside each element and a continuous normal component along Γ. However, the IPM provides a symmetric bilinear form for the velocity, see equation (11a), whereas the formulation proposed in [7] does not.…”

The local discontinuous Galerkin method for the Oseen equations

“…These results apply immediately to the LDG method considered here and in [12]. Conversely, the results in this paper for the extension of the LDG method to the Oseen equations immediately carry over to the methods of Hansbo and Larson [16] and Toselli [28].…”

“…• Methods closely related to our approach are the mixed DG schemes introduced recently by Hansbo and Larson [16] and Toselli [28] in the context of linear elasticity and Stokes flow; these methods also use completely discontinuous approximations for the velocity and the pressure and, as in the approaches of Baker, Karakashian and Jureidini [4,18], are based on interior penalty techniques to weakly enforce the continuity requirements on the velocity fields. In particular, if polynomials of degree k are used for the velocity and polynomials of degree k − 1 for the pressure, a standard inf-sup condition was proved by Hansbo and Larson in [16] in a broken H 1 -seminorm for the velocity and the L 2 -norm for the pressure, giving rise to optimal error estimates.…”

“…In particular, if polynomials of degree k are used for the velocity and polynomials of degree k − 1 for the pressure, a standard inf-sup condition was proved by Hansbo and Larson in [16] in a broken H 1 -seminorm for the velocity and the L 2 -norm for the pressure, giving rise to optimal error estimates. A more refined analysis was then given by Toselli in [28], also covering hanging nodes and extensions to the hp-version. These results apply immediately to the LDG method considered here and in [12].…”