A finite volume method based on the Crouzeix-Raviart element for elliptic PDE's in two dimensions

Abstract

Abstract. We introduce and analyse a finite volume method for the discretization of elliptic boundary value problems in R 2 . The method is based on nonuniform triangulations with piecewise linear nonconforming spaces. We prove optimal order error estimates in the L 2 -norm and a mesh dependent H 1 -norm.

Finite Volume Methods: Foundation and Analysis

“…Finite volume methods for elliptic boundary value problems have been proposed and analyzed under a variety of different names: box methods, covolume methods, diamond cell methods, integral finite difference methods and finite volume element methods, see Bank and Rose, 1987;Cai, 1991;Suli, 1991;Lazarov, Michev and Vassilevsky, 1996;Viozat et al, 1998;Chatzipantelidis, 1999;Chou and Li, 2000;Hermeline, 2000;Eymard, Galluoet and Herbin, 2000;Ewing, Lin and Lin, 2002 elements in a straightforward way. A piecewise constant test space is then constructed using…”

On the Finite Volume Element Method for Self-Adjoint Parabolic Integrodifferential Equations

“…The following lemmas are proved in [3,7], which give the key feature of the bilinear forms in the FVE method.…”

“…The method can be formulated in the finite difference framework or in the Petrov-Galerkin framework. Usually, the former one is called finite volume method [1], marker and cell (MAC) method [2], or cell-centered method [3], and the latter one is called finite volume element method (FVE) [4][5][6][7][8][9], covolume method [10], or vertex-centered method [11,12]. We refer to the monographs [13,14] for general presentation of these methods.…”

A discretization of the phase mass balance in fractional step algorithms for the drift-flux model

“…We also note that when the diffusion term ∇ · τ(u) reduces to the Laplace operator, its 2D CrouzeixRaviart finite element discretization is identical to the finite volume discretization on the dual mesh consisting of the diamond cells D σ [7,6]. This property readily extends to the Rannacher-Turek element and suggests that the advection term ∇ · (ũ n+1 ⊗ ρ n u n ) in (3.2) be discretized on each edge σ ∈ E int by the term ∑ ε∈E (D σ ) F n ε,σũ n+1 ε , where E (D σ ) is the set of the edges of D σ ,ũ n+1 ε is a centred approximation ofũ n+1 on ε and F n ε,σ = |ε| q n ε · n ε , where q n ε denotes an approximation of the momentum ρ n u n on the edge ε, |ε| is the measure of ε and n ε is the normal to ε outward D σ .…”