Higher Order Unfitted Space-Time Finite Element Methods for Moving Domain Problems
by Fabian Heimann
Date of Examination:2024-12-13
Date of issue:2025-01-13
Advisor:Prof. Dr. Christoph Lehrenfeld
Referee:Prof. Dr. Christoph Lehrenfeld
Referee:Prof. Dr. Gert Lube
Persistent Address:
http://dx.doi.org/10.53846/goediss-11003
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Abstract
English
Finite Element methods are a well-established tool for simulating the behaviour of systems from various applications stemming from physics, engineering, and other sciences. In recent years, Unfitted Finite Element methods have been developed as a framework to solve such problems on complicated domains using fixed non-aligning background meshes and an implicit description of the physical geometry. Moreover, in particular, Higher Order Unfitted Finite Elements were developed to reproduce the beneficial error scaling behaviour of higher-order methods within the Unfitted framework. One realisation of this involves an isoparametric mapping, a function deforming the mesh so that the image of a linear reference configuration under the mapping yields a geometrically higher-order approximation. The present work aims to generalise these findings to the moving domain case. Naturally, such problems involving time-dependent domains are of particular interest as applications for Unfitted Finite Elements, as their solution otherwise involves an in general computationally expensive meshing procedure. Starting with a computational perspective, we first present and investigate a generalisation of the Unfitted isoparametric mapping relying on (temporally) discrete levelset functions and their basis entries. To demonstrate the usefulness of the higher-order geometry approximation, we solve a model convection-diffusion problem on moving domains of up to three spatial dimensions. The discretisation in time is based on a space-time approach involving time slices, where continuity is imposed following either Discontinuous or Continuous Galerkin approaches. Higher order convergence of arbitrary order in relevant norms is observed. Second, we switch to a mathematical point of view. It turns out that the careful investigation of the relevant properties yields a more in-depth understanding. In the first half of the rigorous mathematical investigation, we establish properties of interest of the isoparametric mapping. Following a structure developed for the merely spatial case, a perfect counterpart of the discrete high-order mapping is introduced. Its boundedness and upper bounds on the difference between ideal and higher order are established in space and time. In the computational study, a transition between areas where the isoparametric mapping should genuinely operate and regions away from the interface, is applied, called the Finite Element blending. It has essentially the width of one element, which will cause potentially high gradients in relevant norms. To avoid this, we introduce an alternative blending mechanism called smooth blending. The mathematical analysis is uniform regarding these two options and the limitations of the finite element blending are reflected in more restrictive assumptions on time step and mesh size. Finally, the mapping involving the smooth blending is investigated also numerically. Third, we follow up on these mathematical results by presenting a numerical analysis of the Space-Time Finite Element discretisation of the convection-diffusion model problem. Overall, the analysis follows the pattern of Strang's First Lemma, so that errors between discrete and exact geometry will appear as mismatches between exact and discrete bilinear and linear form in the later part of the proof. First, inf-sup stability needs to be established. After a time-slice-wise global reintroduction of the method and the careful consideration of an appropriate version of the Ghost penalty stabilisation on meshes subject to a time-dependent deformation, discrete inequalities are established, which demonstrate how the Ghost penalty stabilisation ensures that stability is not lost in the case of small cuts. Taking into consideration a mesh transfer operator, which is necessary if the finite element blending would be used, and a Lagrange multiplier enforcing weakly the conservation of mass with regards to an appropriate discrete candidate, stability is established. Afterwards, Strang's Lemma is applied, and the geometrical error bounds prove helpful for deriving bounds on the according terms. In regards to interpolation, the boundedness of the perfect mapping counterpart is relevant for the application of a standard L^2 interpolator in space and time. Appropriate upper bounds are obtained for the interpolation summand in that way, and finally, a bound on a Ghost penalty remainder summand is derived as well. After taking these results together into the final result, numerical experiments investigate the effect of including the mass conservation imposition. Finally, a small further application of the methodology developed so far is presented, namely the application to a coupled surface-bulk convection-diffusion problem, and a broader conclusion and outlook are given.
Keywords: Finite Element method; Moving domain problem; Numerical Analysis
