Hierarchical local model reduction for elliptic problems: A domain decomposition approach

Some engineering applications, for instance those 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, which we call 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 in several test cases.