An iterated quasi-asymptotic-preserving hybrid discontinuous Galerkin method for highly anisotropic diffusion problems
Résumé
The paper proposes a quasi-asymptotic-preserving (QAP) scheme based on a hybrid discontinuous Galerkin (HDG) discretization for the resolution of highly anisotropic diffusion problems. The well-posedness of this HDG-QAP scheme is shown for any $\varepsilon>0$ ($1/\varepsilon$ being the anisotropy strength). The direct resolution of the QAP problem discretized using the HDG method being too costly, an algorithm inspired from the Uzawa iteration is proposed, called HDG-UQAP. The iterated solver is equivalent to a modified Uzawa iteration for the limit problem (i.e. with $\varepsilon=0$) and converges for all $\varepsilon\geq0$. Also, numerical tests show that it is asymptotic-preserving, and that it works well even with a severe anisotropy ($\varepsilon=10^{-15}$), the quality of the solutions being unaffected by the anisotropy strength.
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