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978-3-8439-3456-5, Reihe Mathematik

Immanuel Martini Reduced Basis Approximation for Heterogeneous Domain Decomposition Problems

165 Seiten, Dissertation Universität Stuttgart (2017), Softcover, A5

In this thesis a model order reduction method for heterogeneous domain decomposition problems is derived. The focus of the study lies on two specific application cases: the coupling of the Stokes equations and the porous medium equation as an example of a multiphysics simulation and the coupling of the Navier-Stokes equations and the Laplace equation (potential flow) as an example of a zonal approach simulation.

Essential properties of the considered coupled system are a strong coupling, the generalized inf-sup stability concept and a nonlinearity in the second case. Especially, only stationary equations are considered.

The model order reduction approach named Reduced Basis method is successfully applied in scenarios, in which increased demands on efficiency are posed. In developing a reduction procedure, known concepts for nonlinear and inf-sup stable problems on single domains can be applied. The problem structure that is given through the domain decomposition can be exploited in order to achieve a higher efficiency.

A new development represent the so-called Subdomain-Greedy procedures, in which bases on the subdomains are not obtained from global snapshots, but from solutions to local problems. This approach is closely connected to other recent works in this area. The missing boundary condition on the transition between the subdomains is set by so-called interface modes (functions on the interface). The generation of the interface modes is decoupled from the basis generation on the subdomains enabling a flexible choice of the interface modes. The usefulness of the Subdomain-Greedy procedures in practice is illustrated by numerical experiments.

In the case of the zonal approch simulation, not only the reduced basis procedure is investigated, but also the zonal approach itself as an approximation of the global Navier-Stokes equations.

A contribution to the methodology of the Reduced Basis Method in general makes the further development of the kernel interpolation by radial basis functions for the approximation of the (inf-sup) stability constant. It yields a huge acceleration of the online computational time and a good accuracy, as well.