In this paper, we consider the finite element methods for solving second order elliptic and parabolic interface problems in two-dimensional convex polygonal domains. Nearly the same optimal 
$L^2$
 -norm and energy-norm error estimates as for regular problems are obtained when the interfaces are of arbitrary shape but are smooth, though the regularities of the solutions are low on the whole domain. The assumptions on the finite element triangulation are reasonable and practical.

Finite element methods and their convergence for elliptic and parabolic interface problems

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.

/pdf/on-the-computation-of-crystalline-microstructure-2nx6b708rc.pdf

On the computation of crystalline microstructure

The dynamics of phase transitions and hysteresis phenomena in materials with memory are described by a strongly nonlinear coupled system of partial differential equations which, in its generality, can be solved only numerically. Following principles of extended thermodynamics, in this paper we construct a new model for the description of this dynamics based on the Cattaneo-Vernotte law for heat conduction. Models based on the Fourier law follow from this general consideration as special cases. We develop a general procedure for the solution of the resulting systems by their reduction to differential-algebraic systems. Finally, a computational code for the numerical implementation of this procedure is explained in detail, and representative numerical examples are given.

Phase transitions in shape memory alloys with hyperbolic heat conduction and differential-algebraic models

This paper gives a bibliographical review of the finite-element methods (FEMs) applied to the analysis and simulation of smart materials and structures. The bibliography at the end of the paper contains references to papers, conference proceedings and theses/dissertations on the subject that were published between 1986-1997. The following topics are included: smart materials; smart components/structures; smart sensors and actuators; controlled structures technology; and other topics.

https://iopscience.iop.org/article/10.1088/0965-0393/6/3/007/pdf

Smart materials and structures - a finite-element approach: a bibliography (1986-1997)

Smart materials and structures: FEM and BEM simulations a bibliography (1997-1999)

Dans cet article on presente des approximations par elements finis d'un modele de Ginzburg-Landau pour les transitions de phases dans des alliages a memoire de forme. Le systeme non lineaire est discritise en temps par une methode de differences finies et en espace par des elements finis tres simples, lineaires pour la temperature, cubiques de type Hermite pour le deplacement. On obtient des estimations d'erreur pour le schema discretise.

/pdf/finite-element-approximations-of-landau-ginzburg-s-equation-20zgr9t2w0.pdf

Finite element approximations of Landau-Ginzburg's equation model for structural phase transitions in shape memory alloys

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Finite element approximations of Landau-Ginzburg's equation model for structural phase transitions in shape memory alloys