Intrinsic Finite Element Method for Elliptic PDEs on Moving Surfaces

In this thesis, we study Finite Element Methods for elliptic PDEs over moving surface domains, with a particular focus on the accurate formulation of the diffusion equation on time-dependent geometries. We begin by addressing the stationary case and compare two numerical approaches. The first is the Surface Finite Element Method (SFEM), a widely used reference technique that adopts an embedded perspective: differential operators are defined in the three-dimensional space and subsequently projected onto the surface. The second approach, and the main focus of this work, is the Intrinsic Surface Finite Element Method (ISFEM), based entirely on geometric quantities intrinsic to the surface. Since ISFEM can be directly extended to vector and tensor valued equations without introducing additional geometric complexity, this is the approach we choose to treat PDEs over evolving surfaces. To this end, this work introduces the Evolving ISFEM (EISFEM) as a novel extension of ISFEM to evolving geometries. To validate the theoretical formulation of EISFEM, we revisit the Evolving SFEM (ESFEM), which extends SFEM to dynamic surfaces and serves as the benchmark method for this class of problems. A detailed comparison between ESFEM abd EISFEM is carried out at multiple levels. First, we demonstrate the equivalence of both their strong and weak formulations. Next, we show that this equivalence holds up to the construction of the FEM basis. In the last part of this work, we focus on the implementation of both ESFEM and EISFEM. For the ESFEM we reproduced the results displayed in literature. Finally, we tested our novel approach over a variety of examples.