Browsing by Author "Prohl, Andreas"

Now showing 1 - 11 of 11
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    Analysis of a fully discrete finite element method for the phase field model and approximation of its sharp interface limits
    (2001-07) Feng, Xiaobing; Prohl, Andreas
    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 $\varepsilon$, 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 $\frac{1}{\varepsilon}$ only in some lower polynomial order for small $\varepsilon$. The cruxes of the analysis are to establish stability estimates for the discrete solutions, to use a spectrum estimate result of Chen [Comm. PDE, 1371-1395, 1994] 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.
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    Analysis of total variation flow and its finite element approximations
    (2002-07) Feng, Xiaobing; Prohl, Andreas
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    Multiscale resolution in the computation of crystalline microstructure
    (2002-02) Bartels, Soren; Prohl, Andreas
    This paper addresses the numerical approximation of microstructures in crystalline phase transitions without surface energy. It is shown that branching of different variants near interfaces of twinned martensite and simple austenite phases leads to reduced energies in finite element approximations. Such behavior of minimizing deformations is understood for an extended model that involves surface energies. Moreover, the closely related question of the role of different growth conditions of the employed bulk energy is discussed. By explicit construction of discrete deformations in lowest order finite element spaces we prove upper bounds for the energy and thereby clarify the question of the dependence of the convergence rate upon growth conditions and lamination orders. For first order laminates the estimates are optimal.
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    Numerical analysis of the Cahn-Hilliard equation and approximation for the Hele-Shaw problem, Part I: Error analysis under minimum regularities
    (2001-07) Feng, Xiaobing; Prohl, Andreas
    In this first part of a series, we propose and analyze, under minimum regularity assumptions, a semi-discrete (in time) scheme and a fully discrete mixed finite element scheme for the Cahn-Hilliard equation $u_t+\Delta (\varepsilon \Delta u -{\varepsilon}^{-1}f(u))=0$ arising from phase transition in materials science, where $\vepsi$ is a small parameter known as an ``interaction length". The primary goal of this paper is to establish some useful a priori error estimates for the proposed numerical methods, in particular, by focusing on the dependence of the error bounds on $\varepsilon$. Quasi-optimal order error bounds are shown for the semi-discrete and fully discrete schemes under different constraints on the mesh size $h$ and the local time step size $k_m$ of the stretched time grid, and minimum regularity assumptions on the initial function $u_0$ and domain $\Omega$. In particular, all our error bounds depend on $\frac{1}{\varepsilon}$ only in some lower polynomial order for small $\varepsilon$. The cruxes of the analysis are to establish stability estimates for the discrete solutions, to use a spectrum estimate result of Alikakos and Fusco [3] and Chen [15], and to establish a discrete counterpart of it for a linearized Cahn-Hilliard operator to handle the nonlinear term on the stretched time grid. It is this polynomial dependency of the error bounds that paves the way for us to establish convergence of the numerical solution to the solution of the Hele-Shaw (Mullins-Sekerka) problem (as $\varepsilon \searrow 0$) in Part II \cite{XA3} of the series.
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    Numerical analysis of the Cahn-Hilliard equation and approximation for the Hele-Shaw problem, Part II: Error analysis and convergence of the interface
    (2001-07) Feng, Xiaobing; Prohl, Andreas
    In this second part of the series, we focus on approximating the Hele-Shaw problem via the Cahn-Hilliard equation $u_t+\Delta (\varepsilon \Delta u -{\varepsilon}^{-1}f(u))=0$ as $\varepsilon \searrow 0$. 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 Hele-Shaw (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 $L^\infty(L^\infty)$-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.
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    Recent developments in modeling, analysis and numerics of ferromagnetism
    (2004-10) Kruzik, Martin; Prohl, Andreas
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