We consider a system of weakly coupled singularly perturbed convection-diffusion equations with multiple scales. Based on sharp estimates for first order derivatives, Linß [T. Linß, Computing 79 (2007) 23–32.] analyzed the upwind finite-difference method on a Shishkin mesh. We derive such sharp bounds for second order derivatives which show that the coupling generates additional weak layers. Finally, we prove the first robust convergence result for the Galerkin finite element method for this class of problems on modified Shishkin meshes introducing a mesh grading to cope with the weak layers. Numerical experiments support our theory.
DOI : 10.1051/m2an/2015027
Mots-clés : Convection-diffusion, graded mesh, Shishkin mesh, singular perturbation, system of differential equations, uniform convergence
@article{M2AN_2015__49_5_1525_0,
author = {Roos, Hans-G\"org and Schopf, Martin},
title = {Layer structure and the galerkin finite element method for a system of weakly coupled singularly perturbed convection-diffusion equations with multiple scales},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1525--1547},
publisher = {EDP-Sciences},
volume = {49},
number = {5},
year = {2015},
doi = {10.1051/m2an/2015027},
mrnumber = {3423235},
zbl = {1332.65116},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2015027/}
}
TY - JOUR AU - Roos, Hans-Görg AU - Schopf, Martin TI - Layer structure and the galerkin finite element method for a system of weakly coupled singularly perturbed convection-diffusion equations with multiple scales JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2015 SP - 1525 EP - 1547 VL - 49 IS - 5 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2015027/ DO - 10.1051/m2an/2015027 LA - en ID - M2AN_2015__49_5_1525_0 ER -
%0 Journal Article %A Roos, Hans-Görg %A Schopf, Martin %T Layer structure and the galerkin finite element method for a system of weakly coupled singularly perturbed convection-diffusion equations with multiple scales %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2015 %P 1525-1547 %V 49 %N 5 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2015027/ %R 10.1051/m2an/2015027 %G en %F M2AN_2015__49_5_1525_0
Roos, Hans-Görg; Schopf, Martin. Layer structure and the galerkin finite element method for a system of weakly coupled singularly perturbed convection-diffusion equations with multiple scales. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 49 (2015) no. 5, pp. 1525-1547. doi : 10.1051/m2an/2015027. http://www.numdam.org/articles/10.1051/m2an/2015027/
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