Analysis of a projection method for low-order nonconforming finite elements

Abstract

We present a study of the incremental projection method to solve incompressible unsteady Stokes equations based on a low degree non-conforming finite element approximation in space, with, in particular, a piecewise constant approximation for the pressure. The numerical method falls in the class of algebraic projection methods. We provide an error analysis in the case of Dirichlet boundary conditions, which confirms that the splitting error is second-order in time. In addition, we show that pressure artificial boundary conditions are present in the discrete pressure elliptic operator, even if this operator is obtained by a splitting performed at the discrete level; however, these boundary conditions are imposed in the finite volume (weak) sense and the optimal order of approximation in space is still achieved, even for open boundary conditions. Keywords: Incompressible flows, unsteady Stokes problem, projection methods, RannacherTurek finite elements, Crouzeix-Raviart finite elements IntroductionWe consider the time-dependent incompressible Stokes equations, posed on a finite time interval (0, T ) and in an open, connected, bounded domain Ω in R d (d = 2, or 3), which is supposed to be polygonal (d = 2) or polyhedral (d = 3) for the sake of simplicity. The system under consideration reads:where u stands for the (vector-valued) velocity, p for the (scalar) pressure, and f for a (vectorvalued) regular known forcing term. The boundary Γ of Ω is supposed to be split in Γ = Γ D ∪ Γ N , and the measure of Γ D is assumed to be positive. The velocity is prescribed over Γ D while Neumann boundary conditions are imposed over Γ N :This system must be supplemented by the initial condition u = u 0 on {0} × Ω. The vector fields u D , f N and u 0 are given and supposed to be regular. We present in this paper a discretization of System (1.1) with non-conforming low-degree Rannacher-Turek [19] or Crouzeix-Raviart [6] finite elements. The time discretization is performed by an incremental projection method [5,23, 9,24]. Since the pressure is approximated by piecewise constant functions, the projection step must be left as a Darcy system. We thus choose to use a lumped discretization for the time -derivative terms, which allows us to obtain the elliptic problem for the pressure by an explicit algebraic process. Extended to variabledensity Navier-Stokes equations, this scheme is used in the open-source ISIS code [14] developed at IRSN for the computation of low-Mach-number turbulent reactive flows, and extensively used for simulation of fires.Our results are twofold. First, we are able to lay down the scheme in a variational setting, with mesh-dependent inner-products, operators and norms, which allows us to adapt for the problem at hand the error analysis performed in the semi-discrete time setting [20, 11] or for conforming elements [10]; we thus obtain, for homogeneous Dirichlet boundary conditions, a second-order estimate (with respect to the time step) for the splitting error. Second, we derive an explicit expression for the d...

Pressure correction staggered schemes for barotropic one-phase and two-phase flows

“…It is indeed the case for the present scheme, even if it is derived by an algebraic splitting (i.e. by discretizing first the equations up to obtain an implicit fully discrete scheme and then splitting in time, instead of first writing a split time semi-discrete algorithm with (artificial) boundary conditions explicitly stated at each step): we show in [7] that the elliptic problem solved at the correction step for the pressure increment takes the form of a finite volume diffusion problem, with homogeneous Neumann conditions at the boundary where the velocity is prescribed and homogeneous Dirichlet conditions when the velocity is free. In the present case, it means in particular that the pressure suffers from a numerical boundary condition at the outlet section which tends to fix it at the initial value.…”

“…Note, however, that this boundary condition is only prescribed in the "finite volume way" (i.e. through the expression of the flux), which may be seen as a penalization process with a δt/h coefficient, so this condition is relaxed when this latter ratio is small [7]. We observe in Figure 22 that this outlet condition indeed generates a pressure boundary layer.…”

An unconditionally stable staggered pressure correction scheme for the compressible Navier-Stokes equations

“…-a rectangle (d = 2) or a rectangular parallelepiped (d = 3); in this case, we denote by x K the mass center of K; -a simplex, the circumcenter x K of which is located inside K. (10) This condition implies that, for each neighboring control volumes K and L, the segment [x K , x L ] is orthogonal to the face K|L separating K from L, even when, in two space dimensions, one cell is a rectangle and the other one a triangle (we recall that, in three space dimensions, the two types of cells cannot be mixed). For each internal face σ = K|L, we denote by…”

“…Pressure correction schemes are known to generate spurious boundary conditions for the pressure, which, for the discretization used here, are implicit in the pressure elliptic operator in the correction step (see [10,Section 2.3] for a discussion on this topic, with the same space discretization as here but for the toy problem is set to µ = 0.001, which roughly corresponds to a fifth of the numerical viscosity introduced by the classical upwinding µ upw ρ |u| h/2 of the convection term.…”

“…Pressure correction schemes are known to generate spurious boundary conditions for the pressure, which, for the discretization used here, are implicit in the pressure elliptic operator in the correction step (see [10,Section 2.3] for a discussion on this topic, with the same space discretization as here but for the toy problem of the time-dependent incompressible Stokes equations, and Appendix C of the present paper). For a free outlet boundary (as for a Neuman condition), the artificial boundary condition is a non-homogeneous Dirichlet boundary condition for the pressure, with the prescribed value p ext corresponding to the external pressure used in the gradient approximation at the boundary faces.…”