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.
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DOI : 10.1051/m2an/2016042
Mots clés : Convection-diffusion equation, combined finite element – finite volume method, Crouzeix–Raviart finite elements, barycentric finite volumes, upwind method, stability
@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://www.numdam.org/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://www.numdam.org/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://www.numdam.org/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. http://www.numdam.org/articles/10.1051/m2an/2016042/
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