In this paper we focus on melting and solidification processes described by phase-field models and obtain rigorous estimates for such processes. These estimates are derived in Section 2 and guarantee the convergence of solutions to non-constant equilibrium patterns. The most basic results conclude with the inequality (E2.31). The estimates in the remainder of Section 2 illustrate what obtains if the initial data is progressively more regular and may be omitted on first reading. We also present some interesting numerical simulations which demonstrate the equilibrium structures and the approach of the system to non-constant equilibrium patterns. The novel feature of these calculations is the linking of the small parameter in the system, δ, to the grid spacing, thereby producing solutions with approximate sharp interfaces. Similar ideas have been used by Caginalp and Sokolovsky [5]. A movie of these simulations may be found at http:www.math.cmu.edu/math/people/greenberg.html

We propose a simple numerical method for capturing thesteady state solution of hyperbolic systems with geometricalsource terms. We usethe interface value, rather than the cell-averages, for the source terms that balance the nonlinear convectionat the cell interface, allowing the numerical capturing of the steadystate with a formal high order accuracy. This method applies to Godunovor Roe type upwind methods butrequires no modification of the Riemann solver. Numerical experiments on scalar conservationlaws and the one dimensional shallow water equationsshow much better resolution of the steady state than the conventionalmethod, with almost no new numerical complexity.

This paper deals with the use of wavelets in the framework of the Mortar method.We first review in an abstract framework the theory of the mortar method fornon conforming domain decomposition, and point out some basic assumptionsunder which stability and convergence of such method can be proven. We studythe application of the mortar method in the biorthogonal wavelet framework.In particular we define suitable multiplier spaces for imposing weakcontinuity. Unlike in the classical mortar method, such multiplier spaces arenot a subset of the space of traces of interior functions, but rather oftheir duals.For the resulting method, we provide with an error estimate, which is optimal in thegeometrically conforming case.

We analyze a two species discrete predator-prey model inwhich the prey disperses between two patches of a heterogeneous environmentwith barriers and the mature predator disperses between thepatches with no barrier. By using the discrete dynamical system generated by a monotone, concavemaps for subcommunity of prey, we obtain the subcommunity of preyexists an equilibrium which attracts all positive solutions, andusing the stability trichotomy results on the monotone andcontinuous operator, we obtain some sufficient conditions for thepermanence of species. These results are applied to the modelswith rational growth functions and exponential growth functions.We also present numerical examples to illustrate the dynamiccomplexity of systems.

We consider a nonlinear second order elliptic boundary value problem (BVP)in a bounded domain $\Omega\subset {\mathbb R}^N$ witha nonlocal boundary condition. A Dirichlet BC containing an unknown additive constant,accompanied with a nonlocal (integral) Neumann side condition is prescribed at some boundary part Γn . The rest of the boundary is equipped with Dirichlet or nonlinear Robin type BC. The solution is found via linearization. We design a robust and efficient approximation scheme. Error estimates for the linearization algorithm are derived inL 2(Ω),H 1(Ω) andL ∞(Ω) spaces.

We consider a model for phase separation of a multi-component alloy with non-smooth free energyand a degenerate mobility matrix. In addition to showingwell-posedness and stability bounds for our approximation, we prove convergence in one space dimension. Furthermore an iterative scheme for solving the resulting nonlinear discrete system is analysed. We discuss also how our approximation has to be modified in orderto be applicable to a logarithmic free energy.Finally numerical experiments with three components in one and two space dimensions are presented.

In this article, we consider the initial value problem which is obtained after a space discretization (with space step h) of the equations governing the solidification process of a multicomponent alloy. We propose a numerical scheme to solve numerically this initial value problem. We prove an error estimate which is not affected by the step size h chosen in the space discretization. Consequently, our scheme provides global convergence without any stability condition between h and the time step size τ. Moreover, it is not of excessive algorithmic complexity since it does not require more than one resolution of a linear system at each time step.

The topic of this work is to obtain discrete Sobolev inequalities for piecewise constant functions, and to deduce Lp error estimates on the approximate solutions of convection diffusion equations by finite volume schemes.

Many problems in quantumchemistry deal with the computation of fundamental or excited states of molecules and lead to the resolution of eigenvalue problems. One of the major difficulties in these computations lies in the very largedimension of the systems to be solved. Indeed these eigenfunctions dependon 3n variables where n stands for the number of particles (electrons and/or nucleari) in the molecule. In order to diminish the size of the systems to be solved, the chemists have proposed manyinteresting ideas. Among those stands the adiabatic variable method; we present in this paper a mathematical analysis of thisapproximation and propose, in particular, an a posteriori estimate thatmight allow for verifying the adiabaticity hypothesis that is done on some variables; numerical simulations that support thea posteriori estimators obtained theoretically are also presented.

We present a finite element method to compute guided modes in a stratified medium. The major difficulty to overcome is related to the unboundedness of the stratified medium. Our method is an alternative to the use of artificial boundary conditions and to the use of integral representation formulae. The domain is bounded in such a way we can write the solution on its lateral boundaries in terms of Fourier series. The series is then truncated for the computations over the bounded domain. The problem is scalar and 2-dimensional.

Three non-overlapping domain decomposition methods are proposed for thenumericalapproximation of time-harmonic Maxwell equations with damping (i.e., in a conductor). Foreach method convergence is proved and, for the discrete problem, the rate ofconvergenceof the iterative algorithm is shown to be independent of the number ofdegrees of freedom.