$L^2$-Stability Independent of Diffusion for a Finite Element--Finite Volume Discretization of a Linear Convection-Diffusion Equation

Abstract

We consider a time-dependent and a steady linear convection-diffusion equation. These equations are approximately solved by a combined finite element-finite volume method: the diffusion term is discretized by Crouzeix-Raviart piecewise linear finite elements on a triangular grid, and the convection term by upwind barycentric finite volumes. In the unsteady case, the implicit Euler method is used as time discretization. This scheme is shown to be unconditionally L 2-stable, uniformly with respect to the diffusion coefficient.

L2-stability of a finite element – finite volume discretization of convection-diffusion-reaction equations with nonhomogeneous mixed boundary conditions

“…The article most closely related to our work here and in [15] is reference [3], in which Ayuso and Marini study stability and accuracy of various discontinuous Galerkin approximations of (1.7), (1.8), with discrete convection terms similar to ours. These authors base their theory on condition (1.9) 2 -to our knowledge the only ones doing so previous to [15] in the context of convection-diffusion equations.…”

“…The first two estimates in the lemma are an immediate consequence of (2.7), (3.3) 2 and the CauchySchwarz inequality for sums. As for the third, it follows by modifying the proof of ( [15], Lem. 3.2).…”

“…Actually this term was inserted into the definition of the term d h (t, v h , w h ) because otherwise we could not see how to carry through this proof. Since this term was not present in [15], we elaborate this proof in more detail.…”

“…Our results should be expected to hold in the 3D case as well, although their proof would become more technical. When we adapted the discretization from [15] to the inhomogeneous boundary conditions considered in the work at hand, it was not so clear how to take account of the Robin−Neumann condition in (1.2) and (1.8).…”

“…These authors base their theory on condition (1.9) 2 -to our knowledge the only ones doing so previous to [15] in the context of convection-diffusion equations. However, their stability constant depends exponentially on max φ − min φ ( [3], Lem.…”