Hierarchical local model reduction for elliptic problems I: a domain decomposition approach
Résumé
Some engineering applications, for instance related to fluid dynamics in pipe or channel networks, feature a dominant spatial direction along which the most relevant dynamics develop. Nevertheless, local features of the problem depending on the other directions, that we call \emph{transverse}, can be locally relevant to the whole problem. We propose in the context of elliptic problems such as advection--diffusion--reaction equations, a hierarchical model reduction approach in which a coarse model featuring only the dominant direction dynamics is enriched locally by a fine model that accounts for the transverse variables via an appropriate modal expansion. We introduce a domain decomposition approach allowing us to employ a different number of modal functions in different parts of the domain according to the local complexity of the problem at hand. The methodology is investigated numerically on several test cases.
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