Stability of a combined finite element ‐ finite volume discretization of convection‐diffusion equations

Abstract

We consider a time-dependent and a stationary convection-diffusion equation. These equations are approximated by a combined finite element -finite volume method: the diffusion term is discretized by CrouzeixRaviart piecewise linear finite elements on a triangular grid, and the convection term by upwind barycentric finite volumes. In the nonstationary case, we use an implicit Euler approach for time discretization. This scheme is shown to be L 2 -stable uniformly with respect to the diffusion coefficient. In addition, it turns out that stability is unconditional in the time-dependent case. These results hold if the underlying grid satisfies a condition that is fulfilled, for example, by some structured meshes.

A finite element–finite volume discretization of convection‐diffusion‐reaction equations with nonhomogeneous mixedboundary conditions: Error estimates

“…In such a framework, we could show that our discrete convection term possesses some sort of coercivity property with respect to the norm of . In essence, it is this property which allowed us in to derive ν ‐independent stability estimates for solutions to (1.1)–(1.3) and (1.4), (1.5), respectively, thus generalizing earlier results in (homogeneous Dirichlet boundary conditions; as principal order term in (1.1) and (1.5); b only x ‐dependent) and ( b and B constant vectors; special grids). In the present article, we do not impose (1.10), but we have to accept a term in our error bounds.…”

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

“…In [12], we applied this FE-FV method to problem (1.1), (1.2), using the implicit Euler method as time discretization. Under the assumption that β is constant, we showed that the approximate solution provided by this approach may be estimated in the L 2 -norm against the data, with the constant in this estimate being independent of the diffusion parameter ν.…”

“…It is the aim of the work at hand to establish the stability estimates from [12] without this inconvenient condition. In fact, we will show that even if the grid is only required to be shaperegular (minimum angle condition), an upper bound independent of ν and only involving the data may still be constructed for the L 2 -norm of the approximate solution of (1.1), (1.2) and (1.4), (1.5), respectively, obtained by the FE-FV approach described above, with the implicit Euler method being used as time discretization in the unsteady case.…”

“…In fact, we will show that even if the grid is only required to be shaperegular (minimum angle condition), an upper bound independent of ν and only involving the data may still be constructed for the L 2 -norm of the approximate solution of (1.1), (1.2) and (1.4), (1.5), respectively, obtained by the FE-FV approach described above, with the implicit Euler method being used as time discretization in the unsteady case. As an additional generalization with respect to [12], the function β need no longer be constant. Instead it will only be supposed to satisfy (1.3).…”

“…The key argument allowing us to improve the theory from [12] consists of a weak BV estimate providing an upper bound for a certain weighted variation of our approximate solution. This upper bound involves the L 2 -norm of the solution and of the data.…”