Shibin Dai

On the Efficiency of Adaptive Finite Element Methods for Elliptic Problems with Discontinuous Coefficients

The successful implementation of adaptive finite element methods based on a posteriori error estimates depends on several ingredients: an a posteriori error indicator, a refinement/coarsening strategy, and the choice of various parameters. The objective of the paper is to examine the influence of these factors on the performance of adaptive finite element methods for a model problem: the linear elliptic equation with strongly discontinuous coefficients. We derive a new a posteriori error estimator which depends locally on the oscillations of the coefficients around singular points. Extensive numerical experiments are reported to support our theoretical results and to show the competitive behaviors of the proposed adaptive algorithm.

Adaptive Galerkin Methods with Error Control for a Dynamical Ginzburg--Landau Model in Superconductivity

The time-dependent Ginzburg--Landau model which describes the phase transitions taking place in superconductors is a coupled system of nonlinear parabolic equations. It is discretized semi-implicitly in time and in space via continuous piecewise linear finite elements. A posteriori error estimates are derived for the

Geometric evolution of bilayers under the functionalized Cahn–Hilliard equation

L^\infty L^2

Universal bounds on coarsening rates for mean-field models of phase transitions

norm by studying a dual problem of the linearization of the original system, other than the dual of error equations. Numerical simulations are included which illustrate the reliability of the estimators and the flexibility of the proposed adaptive method.

Motion of interfaces governed by the Cahn-Hilliard equation with highly disparate diffusion mobility

We use a multi-scale analysis to derive a sharp interface limit for the dynamics of bilayer structures of the functionalized Cahn–Hilliard equation. In contrast to analysis based on single-layer interfaces, we show that the Stefan and Mullins–Sekerka problems derived for the evolution of single-layer interfaces for the Cahn–Hilliard equation are trivial in this context, and the sharp interface limit yields a quenched mean-curvature-driven normal velocity at O(ϵ−1), whereas on the longer O(ϵ−2) time scale, it leads to a total surface area preserving Willmore flow. In particular, for space dimension n=2, the constrained Willmore flow drives collections of spherically symmetric vesicles to a common radius, whereas for n=3, the radii are constant, and for n≥4 the largest vesicle dominates.

An upper bound on the coarsening rate for mushy zones in a phase-field model

We prove one-sided universal bounds on coarsening rates for two kinds of mean-field models of phase transitions, one with a coarsening rate

Crossover in coarsening rates for the monopole approximation of the Mullins-Sekerka model with kinetic drag

l \sim t^{1/3}

On the Shortening Rate of Collections of Plane Convex Curves by the Area-Preserving Mean Curvature Flow

and the other with

Computational studies of coarsening rates for the Cahn-Hilliard equation with phase-dependent diffusion mobility

l\sim t^{1/2}

Coarsening Mechanism for Systems Governed by the Cahn--Hilliard Equation with Degenerate Diffusion Mobility

. Here l is a characteristic length scale. These bounds are both proved by following a strategy developed by Kohn and Otto [ Comm. Math. Phys., 229 (2002), pp. 375--395]. The