Anisotropic variational mesh adaptation for embedded finite element methods

Embedded or immersed boundary methods (IBM) are powerful mesh-based techniques that permit to solve partial differential equations (PDEs) in complex geometries circumventing the need of generating a mesh that fits the domain boundary, which is indeed very difficult and has been the main bottleneck of the simulation pipeline for decades. Embedded methods exploit a simple background mesh that covers the domain and the difficulties are (1) the imposition of boundary conditions, (2) the ill-conditioning generated by poorly intersected elements and (3) the lack of resolution required in boundary layers. Whereas several methods are available in the literature to address the first two difficulties, the third one still deserves attention. Meshless methods, Chimera grids or adaptive h or p-refinement strategies have been proposed but none of them include alignment techniques.