A discontinuous Galerkin pressure correction scheme for the incompressible Navier-Stokes equations: stability and convergence

Abstract

The numerical simulation of the incompressible Navier-Stokes equations presents a challenging computational task primarily because of two reasons: (a) the coupling of the velocity and pressure by the incompressibility constraint and (b) the nonlinearity of the convection term [14, 18]. The development of splitting schemes aims to overcome these difficulties by decoupling the nonlinearity in the convection term from the pressure term. For an overview of such methods, we refer to the works of Glowinski [15] and of Guermond, Minev, and Shen [18]. In this paper, we will focus on pressure correction schemes. The basic idea of a non-incremental pressure correction scheme in time was first proposed by Chorin and Temam [5, 28]. This scheme was subsequently modified by several mathematicians leading to two major variations: (1) the incremental scheme where a previous value of the pressure gradient is added [16,30] and (2) the rotational scheme where the non-physical boundary condition for the pressure is corrected by using the rotational form of the Laplacian [29]. The main contribution of our work is the theoretical analysis of a discontinuous Galerkin (dG) discretization of the pressure correction approach. We derive stability and a priori error bounds on a family of regular meshes. The discrete velocities are approximated by discontinuous piecewise polynomials of degree k1 and the discrete potential and pressure by polynomials of degree k2. Stability of the solutions is obtained under the constraint k1−1 ≤ k2 ≤ k1+1 whereas the convergence of the scheme is obtained for the case k2 = k1 − 1 because of approximation properties. The proofs are technical and rely on several tools including special lift operators. The semi-discrete error analysis of pressure correction schemes has been extensively studied, see for example the work by Shen and Guermond [21, 27]. The use