This work derives a posteriori error estimates, in two and three dimensions, for the heat equation coupled with Darcy’s law by a nonlinear viscosity depending on the temperature. We introduce two variational formulations and discretize them by finite element methods. We prove optimal a posteriori errors with two types of computable error indicators. The first one is linked to the linearization and the second one to the discretization. Then we prove upper and lower error bounds under regularity assumptions on the solutions. Finally, numerical computations are performed to show the effectiveness of the error indicators.
Mots-clés : Nonlinear Darcy’s equations, heat equation, finite element method, error indicators, residual a posteriori error estimates
@article{M2AN_2019__53_6_2121_0, author = {Dib, S\'er\'ena and Girault, Vivette and Hecht, Fr\'ed\'eric and Sayah, Toni}, title = {A posteriori error estimates for {Darcy{\textquoteright}s} problem coupled with the heat equation}, journal = {ESAIM: Mathematical Modelling and Numerical Analysis }, pages = {2121--2159}, publisher = {EDP-Sciences}, volume = {53}, number = {6}, year = {2019}, doi = {10.1051/m2an/2019049}, mrnumber = {4041519}, zbl = {1434.65226}, language = {en}, url = {http://www.numdam.org/articles/10.1051/m2an/2019049/} }
TY - JOUR AU - Dib, Séréna AU - Girault, Vivette AU - Hecht, Frédéric AU - Sayah, Toni TI - A posteriori error estimates for Darcy’s problem coupled with the heat equation JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2019 SP - 2121 EP - 2159 VL - 53 IS - 6 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2019049/ DO - 10.1051/m2an/2019049 LA - en ID - M2AN_2019__53_6_2121_0 ER -
%0 Journal Article %A Dib, Séréna %A Girault, Vivette %A Hecht, Frédéric %A Sayah, Toni %T A posteriori error estimates for Darcy’s problem coupled with the heat equation %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2019 %P 2121-2159 %V 53 %N 6 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2019049/ %R 10.1051/m2an/2019049 %G en %F M2AN_2019__53_6_2121_0
Dib, Séréna; Girault, Vivette; Hecht, Frédéric; Sayah, Toni. A posteriori error estimates for Darcy’s problem coupled with the heat equation. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 53 (2019) no. 6, pp. 2121-2159. doi : 10.1051/m2an/2019049. http://www.numdam.org/articles/10.1051/m2an/2019049/
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