Optimal Transportation for Adaptive mesh generation and r-refinement

Résumé

In this thesis, we develop fast solvers and adaptive mesh generation pro- cedures based on the Optimal Transport using B-Splines Finite Elements within the Isogeometric Analysis framework. Our approach ensures the construction of a bijection mapping, addressing a major challenge in Iso- geometric Analysis. Initially, we employ standard B-Splines Finite Elements to solve the Monge-Ampère Equation. However, an analysis of this approach reveals significant limitations, particularly when confronted with high variations near the boundary. To overcome this challenge, we derive a new formu- lation utilizing compatible B-Splines discretization based on a discrete DeRham sequence. Furthermore, we introduce new fast solvers using the Fast Diagonalization method to address this issue effectively. We provide various tests and applications to demonstrate the per- formance and efficiency of our novel approach. These include testing our approach using manufactured solutions of the Monge-Ampère equa- tion and employing analytical density functions. Additionally, we apply our solver to elliptic problems and time-dependent problems, addressing challenges such as achieving higher accuracy, accurately capturing sharp interfaces, and effectively reducing oscillations. Finally, the parameterization of nearly all presented geometries relies on CAD-boundary information, employing a novel approach based on optimal transport.

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