Speaker
Description
Many engineering applications, such as fluid–structure interaction and problems involving evolving domains, require efficient numerical methods for the solution of partial differential equations. In these contexts, the geometry of the domain may be complex or change over time. The fictitious domain method is a suitable tool for addressing this class of problems: the main idea is to embed the physical domain into a larger computational domain, where the mesh can be constructed more easily. This strategy avoids the need for boundary-fitted meshes and significantly reduces the complexity of the meshing process. A common strategy in fictitious domain formulations consists of imposing boundary conditions weakly by introducing suitable auxiliary variables supported on the embedded boundary. This approach leads to a mixed formulation, where the main challenge is to design discrete spaces that ensure a uniform inf–sup condition $[1]$. The presentation proposes a method for solving Dirichlet problems based on the introduction of local discrete spaces that satisfy this condition. We employ the use of element-level bubble functions, originally introduced to stabilize finite element computations. Depending on the position of the embedded boundary with respect to the mesh constructed on the larger domain, the finite element local spaces are enriched with two different types of bubble functions. Using a restriction operator as in $[2]$, this enrichment enables the proof of discrete inf–sup stability. Moreover, it allows to relax the requirement introduced in $[2]$ concerning the ratio between the boundary mesh-size and the domain mesh-size.
The presentation will also include classical error estimates and numerical results confirming the theoretical analysis.
$[1]$ Babuška, I (1973). The finite element method with Lagrangian multipliers, Numerische Mathematik, 20(3), 179-192.
$[2]$ Girault V., Glowinski R. (1995). Error analysis of a fictitious domain method applied to a Dirichlet problem, Japan Journal of Industrial and Applied Mathematics, 12(3), 487-514.