Stability and convergence of finite element methods in complex geometries

Abstract

This thesis investigates the stability and error estimates of finite element methods (FEM) for partial differential equations (PDEs) in complex and evolving geometries. It aims to advance the mathematical understanding and numerical analysis of FEM in three challenging settings: time-dependent domains, fluid-structure interaction (FSI), and the maximum-norm stability of isoparametric FEM. The first part addresses the Arbitrary Lagrangian-Eulerian (ALE) FEM for the Stokes equations on moving domains. By establishing optimal L² error bounds of order O(hr+1) for the velocity and O(hr) for the pressure, this work closes a long-standing gap in the literature, where only sub-optimal convergence rates were previously available. A novel duality argument for H⁻¹-error estimate of pressure is developed to obtain optimal estimates for the commutator between the material derivative and the Stokes-Ritz projection. The second part develops and analyzes a fully-discrete loosely coupled scheme for fluid thin-structure interaction problems. A key innovation is the construction and analysis of a coupled non-stationary Ritz projection that satisfies the kinematic interface condition and enables the derivation of optimal L² error estimates. The proposed loosely coupled scheme incorporates stabilization terms to ensure unconditional energy stability and is rigorously shown to achieve optimal convergence in the L² norm. The third part focuses on maximum norm stability of isoparametric FEM in curvilinear polyhedral domains where the geometry cannot be exactly triangulated. This includes the proof of a weak discrete maximum principle and the derivation of optimal maximum-norm error estimates for elliptic equations. For parabolic problems, the thesis establishes the analyticity and maximal Lp-regularity of the semi-discrete FEM and further proves optimal maximum-norm error estimates.