Hybridized globally divergence-free LDG methods. Part I: The Stokes problem

Abstract

We devise and analyze a new local discontinuous Galerkin (LDG) method for the Stokes equations of incompressible fluid flow. This optimally convergent method is obtained by using an LDG method to discretize a vorticity-velocity formulation of the Stokes equations and by applying a new hybridization to the resulting discretization. One of the main features of the hybridized method is that it provides a globally divergence-free approximate velocity without having to construct globally divergence-free finite-dimensional spaces; only elementwise divergence-free basis functions are used. Another important feature is that it has significantly less degrees of freedom than all other LDG methods in the current literature; in particular, the approximation to the pressure is only defined on the faces of the elements. On the other hand, we show that, as expected, the condition number of the Schur-complement matrix for this approximate pressure is of order h −2 in the mesh size h. Finally, we present numerical experiments that confirm the sharpness of our theoretical a priori error estimates.

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

“…The IPM formulation with solenoidal and irrotational spaces proposed here has many points in common with the LDG formulation in compact form presented in [9]. Both consider piecewise polynomial approximations, see Section 4, and a splitting of the approximation space as a sum of solenoidal and irrotational parts, leading to two uncoupled problems: the first for velocities and hybrid pressures, and the second for the computation of pressures in the interior of the elements.…”

“…In fact, the presence of lifting operators in the weak form is an important difference with the IPM method, with consequences in the consistency of the formulation. The IPM formulation is a consistent formulation, in the sense that the solution of the Stokes problem (1) is also a solution of the IPM weak form, whereas the LDG formulation only verifies an approximate orthogonality property, see [9] for details.…”

“…This definition of the jump was previously considered by other authors, see for instance [9], and presents two important advantages: first, it does not depend on a selection of a privileged normal sign on the edges in 2D or faces in 3D, and second, the input and output spaces for the operator coincide, that is, the jump of a scalar is a scalar, the jump of a vector is a vector, etc. Other definitions have been more popular in the past, but do not have these advantages.…”

“…An interesting alternative is weak form in two uncoupled problems: the first one solves for velocity and hybrid pressure and the second one allows the evaluation of pressures in the interior of the elements. The resulting method has many points in common with the LDG formulation in compact form stated in [9]. Namely, both are formulated in terms of piecewise solenoidal velocities and hybrid pressures, the bilinear form is symmetric and positive definite and the pressure in the interior of the elements is computed as a post-process of the solution.…”

“…Nevertheless, different rationales are followed for the LDG and IPM methods, leading to completely different formulations. For instance, one of the most remarkable differences is that the IPM formulation proposed here does not involve lifting operators, which induce an approximate orthogonality property in the LDG formulation [9].…”