Remarks on the links between low‐order DG methods and some finite‐difference schemes for the Stokes problem

Abstract

SUMMARYIn this paper we demonstrate that some well-known finite-difference schemes can be interpreted within the framework of the local discontinuous Galerkin (LDG) methods using the low-order piecewise solenoidal discrete spaces introduced in (SIAM J. Numer. Anal. 1990; 27(6): 1466-1485). In particular, it appears that it is possible to derive the well-known MAC scheme using a first-order Nédélec approximation on rectangular cells. It has been recently interpreted within the framework of the Raviart-Thomas approximation by Kanschat (Int. J. Numer. Meth. Fluids 2007; published online). The two approximations are algebraically equivalent to the MAC scheme, however, they have to be applied on grids that are staggered on a distance h/2 in each direction. This paper also demonstrates that both discretizations allow for the construction of a divergence-free basis, which yields a linear system with a 'biharmonic' conditioning. Both this paper and Kanschat (Int. J. Numer. Meth. Fluids 2007; published online) demonstrate that the LDG framework can be used to generalize some popular finite-difference schemes to grids that are not parallel to the coordinate axes or that are unstructured.

Superconvergence analysis of the MAC scheme for the two dimensional stokes problem

“…As already mentioned in the introduction section, the convergence property has been studied by a lot of researchers (see, e.g., ). The goal of this section is two folds: one is to provide a convergence analysis under the framework of the Petrov‐Galerkin method and the other is to do some necessary preparations for the superconvergence analysis in the next section.…”

“…Therefore, we follow the basic idea of to analyze the properties of these two schemes. It is worth mentioning that along this direction, the stability and convergence of the MAC scheme have been investigated before (see, e.g., ), our main concern in this article is on the analysis of the superconvergence property of the MAC scheme.…”

“…The mathematical theory of the MAC has been heavily investigated in the literature. For instance, the first order convergence property of the MAC scheme has been shown by Nicolaides and his collaborators by regarding the MAC scheme as a covolume method [5,6], by Girault and Lopez [3], Han and Wu [4] by regarding the MAC scheme as a mixed finite element method, by Kanschat [7] and Minev [8] by regarding the MAC scheme as a local discontinuous Galerkin method.…”

Convergence Analysis of Triangular MAC Schemes for Two Dimensional Stokes Equations

“…Since then, the TMAC scheme has been investigated using the finite volume methods approach [13,21,37], the finite element methods approach [18–20,23,24], and the discountinuous Galerkin (DG) approach [7,14,15,45,46]. The MAC scheme can be interpreted within these approaches when the underlying grids are rectangular [23,26,30,34,37]. …”

Approximating second‐order vector differential operators on distorted meshes in two space dimensions

“…The complementary volume or covolume method is a finite-volume-type formulation that allows the MAC scheme to be extended to general unstructured Delaunay-Voronoi meshes [18][19][20][21][22]. The link with low-order mixed finite elements and discontinuous Galerkin methods is highlighted, respectively, in [23][24][25][26][27][28][29]. Recent works on MAC-like methods may be found in [30,31].…”