We analyze upwind difference methods for strongly degenerate convection-diffusion equations in several spatial dimensions. We prove that the local -error between the exact and numerical solutions is , where is the spatial dimension and 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.
Keywords: Degenerate convection-diffusion equations, entropy conditions, finite difference methods, error estimates
@article{M2AN_2016__50_2_499_0, author = {Karlsen, Kenneth Hvistendahl and Risebro, Nils Henrik and Storr{\o}sten, Erlend Briseid}, title = {On the convergence rate of finite difference methods for degenerate convection-diffusion equations in several space dimensions}, journal = {ESAIM: Mathematical Modelling and Numerical Analysis }, pages = {499--539}, publisher = {EDP-Sciences}, volume = {50}, number = {2}, year = {2016}, doi = {10.1051/m2an/2015057}, mrnumber = {3482553}, zbl = {1342.65182}, language = {en}, url = {http://www.numdam.org/articles/10.1051/m2an/2015057/} }
TY - JOUR AU - Karlsen, Kenneth Hvistendahl AU - Risebro, Nils Henrik AU - Storrøsten, Erlend Briseid TI - On the convergence rate of finite difference methods for degenerate convection-diffusion equations in several space dimensions JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2016 SP - 499 EP - 539 VL - 50 IS - 2 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2015057/ DO - 10.1051/m2an/2015057 LA - en ID - M2AN_2016__50_2_499_0 ER -
%0 Journal Article %A Karlsen, Kenneth Hvistendahl %A Risebro, Nils Henrik %A Storrøsten, Erlend Briseid %T On the convergence rate of finite difference methods for degenerate convection-diffusion equations in several space dimensions %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2016 %P 499-539 %V 50 %N 2 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2015057/ %R 10.1051/m2an/2015057 %G en %F M2AN_2016__50_2_499_0
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 , Volume 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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