On the convergence rate of finite difference methods for degenerate convection-diffusion equations in several space dimensions
Karlsen, Kenneth Hvistendahl1 ; Risebro, Nils Henrik1 ; Storrøsten, Erlend Briseid1
1 Department of Mathematics, University of Oslo, P.O. Box 1053, Blindern, 0316 Oslo, Norway
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 50 (2016) no. 2, pp. 499-539.

We analyze upwind difference methods for strongly degenerate convection-diffusion equations in several spatial dimensions. We prove that the local L 1 -error between the exact and numerical solutions is 𝒪 ( Δ x 2 / ( 19 + d ) ) , where d is the spatial dimension and Δx is the grid size. The error estimate is robust with respect to vanishing diffusion effects. The proof makes effective use of specific kinetic formulations of the difference method and the convection-diffusion equation. This paper is a continuation of [K.H. Karlsen, N.H. Risebro E.B. Storrøsten, Math. Comput. 83 (2014) 2717–2762], in which the one-dimensional case was examined using the Kružkov−Carrillo entropy framework.

DOI : 10.1051/m2an/2015057
Classification : 65M06, 65M15, 35K65, 35L65
Mots-clés : Degenerate convection-diffusion equations, entropy conditions, finite difference methods, error estimates
Karlsen, Kenneth Hvistendahl 1 ; Risebro, Nils Henrik 1 ; Storrøsten, Erlend Briseid 1

1 Department of Mathematics, University of Oslo, P.O. Box 1053, Blindern, 0316 Oslo, Norway 
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Karlsen, Kenneth Hvistendahl; Risebro, Nils Henrik; Storrøsten, Erlend Briseid. On the convergence rate of finite difference methods for degenerate convection-diffusion equations in several space dimensions. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 50 (2016) no. 2, pp. 499-539. doi : 10.1051/m2an/2015057. http://www.numdam.org/articles/10.1051/m2an/2015057/

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