Space-time trace finite element methods for partial differential equations on evolving surfaces

• Raum-Zeit-Spur-Finite-Elemente-Methoden für Partielle Differentialgleichungen auf sich bewegenden Oberflächen

Sass, Hauke; Reusken, Arnold (Thesis advisor); Lehrenfeld, Christoph (Thesis advisor)

Aachen : RWTH Aachen University (2022)
Dissertation / PhD Thesis

Dissertation, RWTH Aachen University, 2022

Abstract

In the present thesis, we consider a scalar convection-diffusion equation posed on an evolving surface that models the surfactant concentration adsorbed at an interface in a multiphase flow system. Based on this parabolic model problem, we develop and study efficient and highly accurate finite element methods that can be used for this model problem and a much larger class of surface partial differential equations in various settings. We formulate our surface equations in terms of surface differential operators in Cartesian coordinates, which is convenient for numerical purposes, especially because we prefer an Eulerian description of the geometry. We consider a variational problem that uses infinite-dimensional function spaces which are discontinuous in time. This is a good starting point for a space-time finite element discretisation that allows for an efficient time stepping implementation. Based on this space-time weak formulation, we present fully discrete Eulerian finite element methods that use discontinuous-in-time and continuous-in-space finite elements. A space-time variant of the higher order parametric trace finite element method is considered. The zero level of the approximation of a space-time level set function defines a Lipschitz continuous space-time manifold that piecewise linearly approximates the evolving surface. We work with a computable parametric mapping whose image defines a higher order approximation of the space-time manifold. The finite elements on a volumetric fixed background mesh are used to define test and trial spaces. We include a space-time variant of the volume normal gradient stabilisation in the bilinear form. It turns out that this stabilisation is crucial for the analysis and optimal order convergence in practice. We restrict ourselves to a linear setting and give a rigorous analysis of two variants of these newly introduced finite element methods. Since the approximation of the space-time surface is piecewise smooth but globally Lipschitz continuous only, the analysis poses several additional issues that are not present in a setting with globally smooth geometry. We derive several novel estimates to deal with non-smooth perturbation terms arising from partial integration on the discrete space-time surface in the analysis. For two methods, one being non-symmetric and one being antisymmetric with respect to the discrete material derivative, we obtain well-posedness and optimal order error bounds in a suitable energy norm. We validate our error analysis with numerical experiments, implementing the corresponding methods in the finite element package Netgen\NGSolve\ngsxfem. Due to the unfitted space-time framework, our methods are very robust concerning strong deformations and topological singularities in the geometry. We illustrate this with various adequate experiments. Further numerical tests show that our higher order finite element methods also perform excellently.