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
Keywords: Convection-diffusion, graded mesh, Shishkin mesh, singular perturbation, system of differential equations, uniform convergence
Roos, Hans-Görg 1 ; Schopf, Martin 1
@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},
year = {2015},
publisher = {EDP Sciences},
volume = {49},
number = {5},
doi = {10.1051/m2an/2015027},
mrnumber = {3423235},
zbl = {1332.65116},
language = {en},
url = {https://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 - https://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 https://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
and , Higher order finite element approximation of systems of convection-diffusion-reaction equations with small diffusion. J. Comput. Appl. Math. 246 (2013) 52–64. | MR | Zbl | DOI
, Parameter-uniform finite difference scheme for a system of coupled singularly perturbed convection-diffusion equations. J. Syst. Sci. Complex. 18 (2005) 498–510. | MR | Zbl
, Parameter-uniform finite difference scheme for a system of coupled singularly perturbed convection-diffusion equations. Int. J. Comput. Math. 82 (2005) 177–192. | MR | Zbl | DOI
and , An improved uniformly convergent scheme in space for 1D parabolic reaction-diffusion systems. Appl. Math. Comput. 243 (2014) 57–73. | MR | Zbl
and , Finite element approximation of convection-diffusion problems using graded meshes. Appl. Numer. Math. 56 (2006) 1314–1325. | MR | Zbl | DOI
and , Global pointwise estimates for Green’s matrix of second order elliptic systems. J. Differ. Eqs. 249 (2010) 2643–2662. | MR | Zbl | DOI
and , Analysis of some difference approximations for a singular perturbation problem without turning points. Math. Comput. 32 (1978) 1025–1039. | MR | Zbl | DOI
, Analysis of an upwind finite-difference scheme for a system of coupled singularly perturbed convection-diffusion equations. Computing 79 (2007) 23–32. | MR | Zbl | DOI
, Analysis of a FEM for a coupled system of singularly perturbed reaction-diffusion equations. Numer. Algor. 50 (2009) 283–291. | MR | Zbl | DOI
T. Linß, Layer-Adapted Meshes for Reaction-Convection-Diffusion Problems, in vol. 1985 of Lect. Notes Math. Springer, Berlin (2010). | MR | Zbl
and , A finite element analysis of a coupled system of singularly perturbed reaction-diffusion equations. Appl. Math. Comput. 148 (2004) 869–880. | MR | Zbl
and , Numerical solution of systems of singularly perturbed differential equations. Comput. Meth. Appl. Math. 9 (2009) 165–191. | MR | Zbl | DOI
and , A uniform convergent numerical method for a coupled system of two singularly perturbed linear reaction-diffusion problems. IMA J. Numer. Anal. 23 (2003) 627–644. | MR | Zbl | DOI
, E. O’Riordan and G. I. Shishkin, A numerical method for a system of singularly perturbed reaction-diffusion equations. J. Comput. Appl. Math. 145 (2002) 151–166. | MR | Zbl | DOI
, and , Analytic regularity for a singularly perturbed system of reaction-diffusion equations with multiple scales. Adv. Comput. Math. 39 (2013) 367–394. | MR | Zbl | DOI
H.-G. Roos, Robust numerical methods for singularly perturbed differential equations: a survey covering 2008–2012. ISRN Appl. Math. (2012). Doi: . | DOI | MR
and , Numerical analysis of a system of singularly perturbed convection-diffusion equations related to optimal control. Numer. Math.: Theory Meth. Appl. 4 (2011) 562–575. | MR | Zbl
H.-G. Roos, M. Stynes and L. Tobiska, Robust Numerical Methods for Singularly Perturbed Differential Equations. Springer, Berlin (2008). | Zbl
, Grid approximations of singularly perturbed systems for parabolic convection-diffusion equations with counterflow. Sib. Zh. Vychisl. Mat. 1 (1998) 281–297. | MR | Zbl
, Approximation of elliptic convection-diffusion equations with parabolic boundary layers. Zh. Vychisl. Mat. Mat. Fiz. 40 (2000) 1648–1661. | MR | Zbl
and , A parameter uniform numerical method for a system of singularly perturbed convection-diffusion equations with discontinuous convection coefficients. Int. J. Comput. Math. 87 (2010) 1374–1388. | MR | Zbl | DOI
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