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We consider a time-dependent and a steady linear convection-diffusion-reaction equation whose coefficients are nonconstant. Boundary conditions are mixed (Dirichlet and Robin−Neumann) and nonhomogeneous. Both the unsteady and the steady problem are approximately solved by a combined finite element – finite volume method: the diffusion term is discretized by Crouzeix−Raviart piecewise linear finite elements on a triangular grid, and the convection term by upwind barycentric finite volumes. In the unsteady case, the implicit Euler method is used as time discretization. This scheme is shown to be unconditionally -stable, uniformly with respect to diffusion, except if the Robin−Neumann boundary condition is inhomogeneous and the convective velocity is tangential at some points of the Robin−Neumann boundary. In that case, a negative power of the diffusion coefficient arises. As is shown by a counterexample, this exception cannot be avoided.
Deuring, Paul 1 ; Eymard, Robert 2
@article{M2AN_2017__51_3_919_0,
author = {Deuring, Paul and Eymard, Robert},
title = {$L^{2}$-stability of a finite element {\textendash} finite volume discretization of convection-diffusion-reaction equations with nonhomogeneous mixed boundary conditions},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {919--947},
publisher = {EDP-Sciences},
volume = {51},
number = {3},
year = {2017},
doi = {10.1051/m2an/2016042},
mrnumber = {3666651},
zbl = {1371.65096},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/m2an/2016042/}
}
TY - JOUR
AU - Deuring, Paul
AU - Eymard, Robert
TI - $L^{2}$-stability of a finite element – finite volume discretization of convection-diffusion-reaction equations with nonhomogeneous mixed boundary conditions
JO - ESAIM: Mathematical Modelling and Numerical Analysis
PY - 2017
SP - 919
EP - 947
VL - 51
IS - 3
PB - EDP-Sciences
UR - http://geodesic.mathdoc.fr/articles/10.1051/m2an/2016042/
DO - 10.1051/m2an/2016042
LA - en
ID - M2AN_2017__51_3_919_0
ER -
%0 Journal Article
%A Deuring, Paul
%A Eymard, Robert
%T $L^{2}$-stability of a finite element – finite volume discretization of convection-diffusion-reaction equations with nonhomogeneous mixed boundary conditions
%J ESAIM: Mathematical Modelling and Numerical Analysis
%D 2017
%P 919-947
%V 51
%N 3
%I EDP-Sciences
%U http://geodesic.mathdoc.fr/articles/10.1051/m2an/2016042/
%R 10.1051/m2an/2016042
%G en
%F M2AN_2017__51_3_919_0
Deuring, Paul; Eymard, Robert. $L^{2}$-stability of a finite element – finite volume discretization of convection-diffusion-reaction equations with nonhomogeneous mixed boundary conditions. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 3, pp. 919-947. doi: 10.1051/m2an/2016042
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