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.
Highlights
We consider the convection-diffusion equation∂tu − ν ∆u + β · ∇u = g in Ω × (0, T ), (1.1)supplemented by the initial and boundary conditions u(x, 0) = u(0)(x) for x ∈ Ω, u | ∂Ω × (0, T ) = 0, (1.2)respectively
When we compare our theory with what is available in existing literature, we look for stability estimates for discretizations based on conventional grids and pertaining to scalar convection-diffusion equations whose structure is similar to that of (1.1) or (1.4) and which do not satisfy (1.7)
Constants independent of ν constitute the best possible case one might hope for, the type of theory developed in [38] and its predecessor papers has drawbacks. It requires periodic boundary conditions instead of Dirichlet ones, it assumes Ω to be a rectangle and the grid to be structured according to this geometry, and it relies on Gronwall’s inequality in a way which leads to exponential dependence of stability and error bounds on the W 1,∞-norm of the convective velocity
Summary
Constants independent of ν constitute the best possible case one might hope for, the type of theory developed in [38] and its predecessor papers has drawbacks It requires periodic boundary conditions instead of Dirichlet ones, it assumes Ω to be a rectangle and the grid to be structured according to this geometry, and it relies on Gronwall’s inequality in a way which leads to exponential dependence of stability and error bounds on the W 1,∞-norm of the convective velocity. Due to relation (2.1) and because Ki1 and Ki2 have a common side, we obtain diam Kiν ≤ C diam Kiμ for i ∈ J o, ν, μ ∈ {1, 2}
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