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.

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.

DOI : 10.1051/m2an/2019049
Classification : 65N15, 74S05
Mots-clés : Nonlinear Darcy’s equations, heat equation, finite element method, error indicators, residual a posteriori error estimates
Dib, Séréna 1 ; Girault, Vivette 1 ; Hecht, Frédéric 1 ; Sayah, Toni 1

1 
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     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},
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     year = {2019},
     doi = {10.1051/m2an/2019049},
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     url = {http://www.numdam.org/articles/10.1051/m2an/2019049/}
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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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