Convergence analysis of a locally stabilized collocated finite volume scheme for incompressible flows

Abstract

Abstract. We present and analyse in this paper a novel cell-centered collocated finite volume scheme for incompressible flows. Its definition involves a partition of the set of control volumes; each element of this partition is called a cluster and consists in a few neighbouring control volumes. Under a simple geometrical assumption for the clusters, we obtain that the pair of discrete spaces associating the classical cell-centered approximation for the velocities and cluster-wide constant pressures is inf-sup stable; in addition, we prove that a stabilization involving pressure jumps only across the internal edges of the clusters yields a stable scheme with the usual collocated discretization (i.e., in particular, with control-volume-wide constant pressures), for the Stokes and the Navier-Stokes problem. An analysis of this stabilized scheme yields the existence of the discrete solution (and uniqueness for the Stokes problem). The convergence of the approximate solution toward the solution to the continuous problem as the mesh size tends to zero is proven, provided, in particular, that the approximation of the mass balance flux is second order accurate; this condition imposes some geometrical conditions on the mesh. Under the same assumption, an error analysis is provided for the Stokes problem: it yields first-order estimates in energy norms. Numerical experiments confirm the theory and show, in addition, a second order convergence for the velocity in a discrete L 2 norm.Mathematics Subject Classification. 65N12, 65N15, 65N30, 76D05, 76D07, 76M25.

Optimal estimates on stabilized finite volume methods for the incompressible Navier–Stokes model in three dimensions

“…Therefore, optimal estimates of finite volume methods are difficult to analyze for the nonlinear Navier-Stokes equations. For more detail, the reader can consult the original papers [1,3,4,6,7,8,10,11,12,15,16,17,18].…”

“…For instance, the reader can refer to the researches of Temam in [39], of V. Girault in [19,5], of He et al...... in [21,22,20,32,31,30], of Heywood et al in [23], of Hill in [24], of Thomee et al in [40], of Shen et al in [38]. In contrast, there are few works on numerical analysis of finite volume methods [1,3,6,8,10,11,16,17,18,35,42]. The steady and non-steady cases have been studied by early work in [26,33,28,29,41,7,27].…”

Analysis of a fractional-step scheme for the P $$_1$$ 1 radiative diffusion model

“…Since, by assumption, u 0 ∈ H 1 0 (Ω), the discrete H 1 norm of u 0 , u 0 1,M is bounded by c u 0 H 1 (Ω) where the real number c only depends on Ω and, in a decreasing way, on the parameter θ M characterizing the regularity of the mesh (see e.g. Lemma 3.3 in [9]). Together with the preceding relation, this provides the control of the first term in (12).…”

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