Adaptive Isogeometric Analysis using optimal transport and their fast solvers

The use of adaptive mesh methods is fundamental to the numerical solution of systems of partial differential equations that involve large solution variations or with different scales. Adaptive mesh methods are also a major tool for solving problems with high anisotropy, such those encountered in Computational Plasmas Physics. Grid Adaptation and moving meshes are used in different fields and thelitterature is quite rich [11, 34, 27, 35, 31, 23, 22, 30, 29, 28, 39, 9]. They are also used to generate Anisotropic meshes [41, 44, 25, 26, 24]. The proposed methods in this work aim to construct a one-to-one mapping F that maps a logical domain (patch, computational domain) with the physical domain asshown in Fig. 1. The function F is constructed using B-splines or NURBS surfaces which are widely used in the Computer Aided Design (CAD) community. The use of these tools in numerical simulations was made popular thanks to the introdution of the IsoGeometric Analysis paradigm by Hughes [37]. Because of the geometric interpretation of the control points, B-spline curves and surfaces have become very popular in CAD. We are interested in these features as they allow us to construct a set of mappings, where each of them will map the unit square onto a sub-domain of the physical domain. Using the geometric propertiesof B-splines curves and surfaces, it is easy to stick these mappings together, in order to have a global C1 or even C2 mapping. Local regularity of each mapping is ensured by construction.