Immersed Divergence-Conforming Finite Element Spaces

Abstract

Computational Fluid Dynamics (CFD) offers numerous benefits, notably the ability to study flows that are challenging or costly to investigate using experiments. A central challenge in CFD lies in simulating fluid flow around complex geometries. Additionally, the governing equations follow conservation laws. This thesis aims to establish the foundation for constructing Immersed divergence-conforming finite element spaces that address both challenges. To tackle the first issue, an immersed method is considered. Instead of generating a mesh that conforms to the object where the flow occurs, the approach involves placing the object within a predefined mesh. However, this introduces new challenges, the most significant of which is the ill-conditioning of the associated linear system. The condition number depends on the location of the immersed object and can lead to an extremely large condition number. The second challenge is resolved by discretizing the problem in a way that preserves essential topological and homological structures at a discrete level, utilizing the principles of finite element exterior calculus. Within this thesis, a structure-preserving subcomplex is developed for the de Rham complex in 1D, and its accuracy is validated through numerical experimentation. Optimal convergence is achieved, the discrete inf-sup test is satisfied, and the resulting linear system's conditioning remains unaffected by the placement of the immersed object. However, the constructed structure-preserving subcomplex for the de Rham complex in 2D, known as the immersed divergence-conforming finite element spaces, does not exhibit convergence. For future research, I recommend focusing on simpler vertical and skewed cuts before exploring more intricate immersed shapes. These simpler cuts involve straightforward choices for edge basis functions/1-form basis functions that are relevant. Additionally, if the outcomes remain unsatisfactory, considering a global approach instead of a local approach could be worthwhile.