25 research outputs found
Well-posedness and longtime behavior for the modified phase-field crystal equation
We consider a modification of the so-called phase-field crystal (PFC)
equation introduced by K.R. Elder et al. This variant has recently been
proposed by P. Stefanovic et al. to distinguish between elastic relaxation and
diffusion time scales. It consists of adding an inertial term (i.e. a
second-order time derivative) into the PFC equation. The mathematical analysis
of the resulting equation is more challenging with respect to the PFC equation,
even at the well-posedness level. Moreover, its solutions do not regularize in
finite time as in the case of PFC equation. Here we analyze the modified PFC
(MPFC) equation endowed with periodic boundary conditions. We first prove the
existence and uniqueness of a solution with initial data in a bounded energy
space. This solution satisfies some uniform dissipative estimates which allow
us to study the global longtime behavior of the corresponding dynamical system.
In particular, we establish the existence of an exponential attractor. Then we
demonstrate that any trajectory originating from the bounded energy phase space
does converge to a unique equilibrium. This is done by means of a suitable
version of the {\L}ojasiewicz-Simon inequality. A convergence rate estimate is
also given
A partition of unity method for the displacement obstacle problem of clamped Kirchhoff plates
A partition of unity method for the displacement obstacle problem of clamped Kirchhoff plates is considered in this paper. We derive optimal error estimates and present numerical results that illustrate the performance of the method. © 2013 Elsevier B.V. All rights reserved
Second-Order Convex Splitting Schemes for Gradient Flows with Ehrlich-Schwoebel Type Energy: Application to Thin Film Epitaxy
We construct unconditionally stable, unconditionally uniquely solvable, and second order accurate (in time) schemes for gradient flows with energy of the form {equation presented} dx. the construction of the schemes involves the appropriate combination and extension of two classical ideas: (i) appropriate convex-concave decomposition of the energy functional and (ii) the secant method. as an application, we derive schemes for epitaxial growth models with slope selection (F(y) = 1/4 (|y| 2 - 1) 2) or without slope selection (F(y) = -1/2 ln(1 + |y| 2)). Two types of unconditionally stable uniquely solvable second-order schemes are presented. the first type inherits the variational structure of the original continuous-in-time gradient flow, while the second type does not preserve the variational structure. We present numerical simulations for the case with slope selection which verify well-known physical scaling laws for the long time coarsening process. © 2012 Society for Industrial and Applied Mathematics