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We consider the enhancement of accuracy, by means of a simple post-processing technique, for finite element approximations to transient hy-perbolic 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… (More)

We consider a class of nonconforming nite element approximations of a simply laminated microstructure which minimizes the nonconvex variational problem for the deformation of martensitic crystals which can undergo either an orthorhombic to monoclinic (double well) or a cubic to tetrag-onal (triple well) transformation. We rst establish a series of error… (More)

We derive a model problem for quasicontinuum approximations that allows a simple , yet insightful, analysis of the optimal-order convergence rate in the continuum limit for both the energy-based quasicontinuum approximation and the quasi-nonlocal quasicontinuum approximation. For simplicity, the analysis is restricted to the case of second-neighbor… (More)

- Bo Li, Mitchell Luskin
- 1998

Martensitic crystals which can undergo a cubic to tetragonal phase transformation have a nonconvex energy density with three symmetry-related, rotationally invariant energy wells. We give estimates for the numerical approximation of a first-order laminate for such martensitic crystals. We give bounds for the L 2 convergence of directional derivatives in the… (More)

- Kaushik Bhattacharya, Bo Li, Mitchell Luskin
- 1999

The simply laminated mi-crostructure in martensitic crystals that undergo a cubic to orthorhombic phase trans-Abstract. We study simply laminated microstructures of a martensitic crystal capable of undergoing a cubic to orthorhombic transformation of type P (432) ! P (222) 0. The free energy density modeling such a crystal is minimized on six energy wells… (More)

The accuracy of atomistic-to-continuum hybrid methods can be guaranteed only for deformations where the lattice configuration is stable for both the atomistic energy and the hybrid energy. For this reason, a sharp stability analysis of atomistic-to-continuum coupling methods is essential for evaluating their capabilities for predicting the formation of… (More)

We propose and analyze a goal-oriented a posteriori error estimator for the atomistic-continuum modeling error in the quasicontinuum method. Based on this error estimator, we develop an algorithm which adaptively determines the atomistic and continuum regions to compute a quantity of interest to within a given tolerance. We apply the algorithm to the… (More)

An analysis is given for a class of nonconforming Lagrange-type finite elements which have been successfully utilized to approximate the solution of a variational problem modeling the deformation of martensitic crystals with microstructure. These elements were first proposed and analyzed in 1992 by Rannacher and Turek for the Stokes equation. Our analysis… (More)

We give an analysis of a continuation algorithm for the numerical solution of the force-based quasicontinuum equations. The approximate solution of the force-based quasicontinuum equations is computed by an iterative method using an energy-based quasicontinuum approximation as the preconditioner. The analysis presented in this paper is used to determine an… (More)

- Bo Li, Mitchell Luskin
- 1999

We give results for the approximation of a laminate with varying volume fractions for multi-well energy minimization problems modeling martensitic crystals that can undergo either an orthorhombic to monoclinic or a cubic to tetragonal transformation. We construct energy minimizing sequences of deformations which satisfy the corresponding boundary condition,… (More)