Mitchell Luskin

On the computation of crystalline microstructure

Microstructure is a feature of crystals with multiple symmetry-related energy-minimizing states. Continuum models have been developed explaining microstructure as the mixture of these symmetry-related states on a fine scale to minimize energy. This article is a review of numerical methods and the numerical analysis for the computation of crystalline microstructure.

On the Smoothing Property of the Galerkin Method for Parabolic Equations

We analyze the Galerkin approximation of the general second-order parabolic initial-boundary value problem. For the second-order continuous time method with initial data only in

Enhanced accuracy by post-processing for finite element methods for hyperbolic equations

L^2

On the Smoothing Property of the Crank-Nicolson Scheme

we prove an

Numerical approximation of the solution of variational problem with a double well potential

L^2

An Optimal Order Error Analysis of the One-Dimensional Quasicontinuum Approximation

error estimate of order

Analysis of the finite element approximation of microstructure in micromagnetics

O({{h^2 } / t})

Atomistic-to-continuum coupling

. Analogous results are shown to hold for the error in negative Sobolev norms and for the time derivative of the error. Our analysis uses only elementary energy techniques and the same techniques are shown to give a simple analysis of the backward Euler time discretization.

Optimal order error estimates for the finite element approximation of the solution of a nonconvex variational problem

We consider the enhancement of accuracy, by means of a simple post-processing technique, for finite element approximations to transient hyperbolic equations. The post-processing is a convolution with a kernel whose support has measure of order one in the case of arbitrary unstructured meshes; if the mesh is locally translation invariant, the support of the kernel is a cube whose edges are of size of the order of Δx only. For example, when polynomials of degree k are used in the discontinuous Galerkin (DG) method, and the exact solution is globally smooth, the DG method is of order k+1/2 in the L2-norm, whereas the post-processed approximation is of order 2k + 1; if the exact solution is in L2 only, in which case no order of convergence is available for the DG method, the post-processed approximation converges with order k + 1/2 in L2(Ω0), where Ω0 is a subdomain over which the exact solution is smooth. Numerical results displaying the sharpness of the estimates are presented.

Construction of inertial manifolds by elliptic regularization

The Crank-Nicolson scheme for discretizing linear parabolic equations converges at the rate of only o(1) in L 2 for initial data in L 2. It is shown that smoothing by adding four backward Euler steps to the scheme improves the convergence rate to 0(k 2/t 2). AMS(MOS): 65M10