A posteriori error estimates for Darcy’s problem coupled with the heat equation

Serena Dib 1, 2, 3 Vivette Girault 1, 2 Frédéric Hecht 1, 4 Toni Sayah 3
4 ALPINES - Algorithms and parallel tools for integrated numerical simulations
INSMI - Institut National des Sciences Mathématiques et de leurs Interactions, Inria de Paris, LJLL (UMR_7598) - Laboratoire Jacques-Louis Lions
Abstract : 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 intro- duce 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 lin- earization 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.
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Serena Dib, Vivette Girault, Frédéric Hecht, Toni Sayah. A posteriori error estimates for Darcy’s problem coupled with the heat equation. ESAIM: Mathematical Modelling and Numerical Analysis, EDP Sciences, 2019, ESAIM: M2AN, 53 (6), pp.2121 - 2159. ⟨10.1051/m2an/2019049 ⟩. ⟨hal-02427095⟩



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