Convergence analysis of domain decomposition methods : Nonlinear elliptic and linear parabolic equations

Domain decomposition methods are widely used tools for solving partial differential equations in parallel. However, despite their long history, there is a lack of rigorous convergence theory for equations with non-symmetric differential operators. This includes both nonlinear elliptic equations and linear parabolic equations. The aim of this thesis is therefore twofold: First, to construct frameworks, based on new Steklov--Poincaré operators, that allow the study of nonoverlapping domain decomposition methods for nonlinear elliptic and linear parabolic equations. Second, to prove convergence of the Robin--Robin method using these frameworks. For the nonlinear elliptic case, this involves studying $L^p$-variants of the Lions--Magenes space. In the parabolic case, we use a variational formulation based on fractional time-regularity. The analysis is performed with weak requirements on the spatial domain, where we only assume that the domains have Lipschitz regularity, and for the solutions to the equations, where we assume that their normal derivatives over the interface is in $L^2(\Gamma)$

Numerical Analysis of Transient Teflon Ablation with a Domain Decomposition Finite Volume Implicit Method on Unstructured Grids

This work investigates numerically the process of Teflon ablation using a finite-volume discretization, implicit time integration and a domain decomposition method in three-dimensions. The interest in Teflon stems from its use in Pulsed Plasma Thrusters and in thermal protection systems for reentry vehicles. The ablation of Teflon is a complex process that involves phase transition, a receding external boundary where the heat flux is applied, an interface between a crystalline and amorphous (gel) phase and a depolymerization reaction which happens on and beneath the ablating surface. The mathematical model used in this work is based on a two-phase model that accounts for the amorphous and crystalline phases as well as the depolymerization of Teflon in the form of an Arrhenius reaction equation. The model accounts also for temperature-dependent material properties, for unsteady heat inputs and boundary conditions in 3D. The model is implemented in 3D domains of arbitrary geometry with a finite volume discretization on unstructured grids. The numerical solution of the transient reaction-diffusion equation coupled with the Arrhenius-based ablation model advances in time using implicit Crank-Nicolson scheme. For each time step the implicit time advancing is decomposed into multiple sub-problems by a domain decomposition method. Each of the sub-problems is solved in parallel by Newton-Krylov non-linear solver. After each implicit time-advancing step, the rate of ablation and the fraction of depolymerized material are updated explicitly with the Arrhenius-based ablation model. After the computation, the surface of ablation front and the melting surface are recovered from the scalar field of fraction of depolymerized material and the fraction of melted material by post-processing. The code is verified against analytical solutions for the heat diffusion problem and the Stefan problem. The code is validated against experimental data of Teflon ablation. The verification and validation demonstrates the ability of the numerical method in simulating three dimensional ablation of Teflon