A Uniformly Accurate Finite Volume Discretization for a Convection-diffusion Problem

Abstract

α1, α2 ) > (0, 0) and b(x) − 12 div a(x) ≥ β > 0 for x = (x, y) ∈ Ω. We assume that a, b and f are smooth. The solution u of (1.1) has exponential boundary layers at the sides x = 1 and y = 1 of Ω. There is a vast literature dealing with numerical methods for convection-diffusion and associated problems; see [13, 15] for a survey. We shall consider an inverse-monotone finite volume discretization on layer-adapted meshes. This scheme was introduced by Baba and Tabata [3] and later generalized by Angermann [1, 2] who also realised that Samarski’s scheme [16] fits into this framework. Although we restrict ourselves to piecewise uniform meshes—the so-called Shishkin meshes [12, 18]—our results can be extended to more general meshes, e.g., the Shishkin-type meshes of [14]; see [21, Chapter 3]. A number of numerical methods on Shishkin meshes have been investigated including finite difference schemes [9, 12, 18], Galerkin FEM [7, 19], the streamline diffusion FEM [11, 20] and upwinded FEM with artificial viscosity stabilization [17]. None of these FEM’s is inverse-monotone on highly anisotropic meshes. In contrast, we shall study an inversemonotone finite volume method for (1.1) in this paper. Typically FVM’s are interpreted as FEM’s with inexact integration and therefore most frequently analysed in a finite element context with convergence established in the L2 norm or in weighted H norms [2, 4, 6, 21]. Here we shall pursue a similar approach, but we study convergence in a discrete meshdependent norm. This norm is stronger than the standard ε-weighted energy norm. An outline of the paper is as follows. In Section 2 we define the upwind FVM, study its stability properties and quote some convergence results. The asymptotic behaviour of the solution of (1.1) is investigated in Section 3. We introduce special piecewise uniform layer-adapted meshes and state our main convergence result. The main ideas of the analysis from [21] are presented in Section 4 for the one-dimensional version of (1.1). Finally, we present results of numerical experiments in Section 5. Notation: C denotes a generic positive constant that is independent of ε and of the mesh. Also, we set gi = g(xi) for any function g ∈ C[0, 1], while ui denotes the ith component of the numerical solution u. Similarly, we shall set gi = g(xi) and gij = g(xi, yj) for g ∈ C(Ω̄).

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Cite this paper

@inproceedings{RoosAUA, title={A Uniformly Accurate Finite Volume Discretization for a Convection-diffusion Problem}, author={Hans G{\"{o}rg Roos} }