A Staggered Discontinuous Galerkin Method for the Stokes System

Abstract

Discontinuous Galerkin (DG) methods are a class of efficient tools for solving fluid flow problems. There are in the literature many greatly successful DG methods. In this paper, a new staggered DG method for the Stokes system is developed and analyzed. The key feature of our method is that the discrete system preserves the structures of the continuous problem, which results from the use of our new staggered DG spaces. This also provides local and global conservation properties, which are desirable for fluid flow applications. The method is based on the first order mixed formulation involving pressure, velocity, and velocity gradient. The velocity and velocity gradient are approximated by polynomials of the same degree while the choice of polynomial degree for pressure is flexible, namely, the approximation degree for pressure can be chosen as either that of velocity or one degree lower than that of velocity. In any case, stability and optimal convergence of the method are proved. Moreover, a superconvergence result with respect to a discrete H 1 -norm for the velocity is proved. Furthermore, a local postprocessing technique is proposed to improve the divergence free property of the velocity approximation and it is proved that the postprocessed velocity retains the original accuracy and is weakly divergence free with respect to pressure test functions. Numerical results are included to validate our theoretical estimates and to present the ability of our method for capturing singular solutions. Introduction.Discontinuous Galerkin (DG) methods are gaining popularity in solving fluid flow problems. For example, in [11], the local DG methods were developed for the Stokes system. These are stabilized mixed methods and a nonstandard inf-sup condition was proved. The methods give more flexibility in the choice of approximation spaces for the velocity and pressure. In particular, experimental results there showed that if polynomials of degree k and k − 1 are used to approximate both velocity and pressure, respectively, then the velocity converges with order k + 1 in L 2 while the pressure converges with order k. When polynomials of the same degree k are used to approximate both velocity and pressure, the numerical solution still converges, but the order of convergence for the pressure does not improve because the L 2 error of the pressure also depends on the energy error of the velocity.Due to their discontinuous nature, DG methods are also well suited for hpadaptivity; see [26] for a study on the mixed hp-DGFEM with the Q k −Q k−1 elements and see also [27] in which corner singularities are treated and the authors' derived ex-

A new staggered DG method for the Brinkman problem robust in the Darcy and Stokes limits

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A-posteriori error analysis for a staggered discontinuous Galerkin discretization of the time-harmonic Maxwell’s equations

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“…The idea was subsequently applied to other problems, such as convection-diffusion equations [12], electromagnetic problems [13,[4][5][6]8], Stokes equations [22] and multiscale wave simulations [7,17]. Regarding the approximation of the time harmonic…”

A Staggered Discontinuous Galerkin Method with Local TV Regularization for the Burgers Equation

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