Andreas Prohl

Analysis of a fully discrete finite element method for the phase field model and approximation of its sharp interface limits

We propose and analyze a fully discrete finite element scheme for the phase field model describing the solidification process in materials science. The primary goal of this paper is to establish some useful a priori error estimates for the proposed numerical method, in particular, by focusing on the dependence of the error bounds on the parameter e, known as the measure of the interface thickness. Optimal order error bounds are shown for the fully discrete scheme under some reasonable constraints on the mesh size h and the time step size k. In particular, it is shown that all error bounds depend on 1 a only in some lower polynomial order for small e. The cruxes of the analysis are to establish stability estimates for the discrete solutions, to use a spectrum estimate result of Chen, and to establish a discrete counterpart of it for a linearized phase field operator to handle the nonlinear effect. Finally, as a nontrivial byproduct, the error estimates are used to establish convergence of the solution of the fully discrete scheme to solutions of the sharp interface limits of the phase field model under different scaling in its coefficients. The sharp interface limits include the classical Stefan problem, the generalized Stefan problems with surface tension and surface kinetics, the motion by mean curvature flow, and the Hele-Shaw model.

Convergence of an Implicit Finite Element Method for the Landau--Lifshitz--Gilbert Equation

The Landau--Lifshitz--Gilbert equation describes the dynamics of ferromagnetism, where strong nonlinearity and nonconvexity are hard to tackle: so far, existing explicit schemes to approximate weak solutions suffer from severe time-step restrictions. In this paper, we propose an implicit fully discrete scheme and verify unconditional convergence.

Numerical analysis of the Cahn-Hilliard equation and approximation for the Hele-Shaw problem

In this second part of the series, we focus on approximating the Hele-Shaw problem via the Cahn-Hilliard equation ut + ∆(e∆u − ef(u)) = 0 as e ց 0. The primary goal of this paper is to establish the convergence of the solution of the fully discrete mixed finite element scheme proposed in [21] to the solution of the Hele-Shaw (Mullins-Sekerka) problem, provided that the HeleShaw (Mullins-Sekerka) problem has a global (in time) classical solution. This is accomplished by establishing some improved a priori solution and error estimates, in particular, an L(L)-error estimate, and making full use of the convergence result of [2]. Like in [20, 21], the cruxes of the analysis are to establish stability estimates for the discrete solutions, use a spectrum estimate result of Alikakos and Fusco [3] and Chen [12], and establish a discrete counterpart of it for a linearized Cahn-Hilliard operator to handle the nonlinear term.

Finite Element Approximations of the Ericksen-Leslie Model for Nematic Liquid Crystal Flow

The Ericksen-Leslie model describes dynamics of low molar-mass nematic liquid crystals, where the spatiotemporal distribution of defects defining texture is represented by the director unit vector field

Numerical analysis of an explicit approximation scheme for the Landau-Lifshitz-Gilbert equation

{\bf d}: \Omega_T \rightarrow \mathbb{S}^2

Constraint preserving implicit finite element discretization of harmonic map flow into spheres

. It consists of the Navier-Stokes equations with an extra viscous stress tensor, and a convective harmonic map heat flow equation to govern the dynamics of the director field. Two fully discrete finite element methods, for a regularized system using the Ginzburg-Landau regularization and for the limiting Ericksen-Leslie model, are proposed, and well-posedness and related discrete energy laws are established. For the regularized model, unconditional convergence of finite element solutions towards weak solutions of the continuum model as well as convergence towards measure-valued solutions of the limiting Ericksen-Leslie model are verified when the mesh parameters and the regularization parameter successively tend to zero. Computational experiments are also presented to show the importance of balancing numerical and regularization parameters, to compare regularized and direct approaches, and to show numerically the finite-time formation, annihilation, and evolution of point defects.

Convergent discretizations for the Nernst–Planck–Poisson system

The Landau-Lifshitz-Gilbert equation describes magnetic behavior in ferromagnetic materials. Construction of numerical strategies to approximate weak solutions for this equation is made difficult by its top order nonlinearity and nonconvex constraint. In this paper, we discuss necessary scaling of numerical parameters and provide a refined convergence result for the scheme first proposed by Alouges and Jaisson (2006). As an application, we numerically study discrete finite time blowup in two dimensions.

Numerical analysis of relaxed micromagnetics by penalised finite elements

Discretization of the harmonic map flow into spheres often uses a penalization or projection strategy, where the first suffers from the proper choice of an additional parameter, and the latter from the lack of a discrete energy law, and restrictive mesh-constraints. We propose an implicit scheme that preserves the sphere constraint at every node, enjoys a discrete energy law, and unconditionally converges to weak solutions of the harmonic map heat flow.

On Pressure Approximation via Projection Methods for Nonstationary Incompressible Navier-Stokes Equations

We propose and compare two classes of convergent finite element based approximations of the nonstationary Nernst–Planck–Poisson equations, whose constructions are motivated from energy versus entropy decay properties for the limiting system. Solutions of both schemes converge to weak solutions of the limiting problem for discretization parameters tending to zero. Our main focus is to study qualitative properties for the different approaches at finite discretization scales, like conservation of mass, non-negativity, discrete maximum principle, decay of discrete energies, and entropies to study long-time asymptotics.

A Convergent Implicit Finite Element Discretization of the Maxwell-Landau-Lifshitz-Gilbert Equation

Summary. Some micromagnetic phenomena in rigid (ferro-)magnetic materials can be modelled by a non-convex minimisation problem. Typically, minimising sequences develop finer and finer oscillations and their weak limits do not attain the infimal energy. Solutions exist in a generalised sense and the observed microstructure can be described in terms of Young measures. A relaxation by convexifying the energy density resolves the essential macroscopic information. The numerical analysis of the relaxed problem faces convex but degenerated energy functionals in a setting similar to mixed finite element formulations. The lowest order conforming finite element schemes appear instable and nonconforming finite element methods are proposed. An a priori and a posteriori error analysis is presented for a penalised version of the side-restriction that the modulus of the magnetic field is bounded pointwise. Residual-based adaptive algorithms are proposed and experimentally shown to be efficient.