In this paper, we present and analyze gradient recovery type a posteriori error estimates for the finite element approximation of elliptic variational inequalities of the second kind. Both reliability and efficiency of the estimates are addressed. Some numerical results are reported, showing the effectiveness of the error estimates in adaptive solution of elliptic variational inequalities of the second kind.

The Nordström-Vlasov system provides an interesting relativistic generalization of the Vlasov-Poisson system in the gravitational case, even though there is no direct physical application. The study of this model will probably lead to a better mathematical understanding of the class of non-linear systems consisting of hyperbolic and transport equations. In this paper it is shown that solutions of the Nordström-Vlasov system converge to solutions of the Vlasov-Poisson system in a pointwise sense as the speed of light tends to infinity, providing a further and rigorous justification of this model as a genuine relativistic generalization of the Vlasov-Poisson system.

The orbital stability of solitary waves has generally been established in Sobolev classes of relatively low order, such as $H^1$. It is shown here that at least for solitary-wave solutions of certain model equations, a sharp form of orbital stability is valid in $L^2$-based Sobolev classes of arbitrarily high order. Our theory includes the classical Korteweg-de Vries equation, the Benjamin- Ono equation and the cubic, nonlinear Schrödinger equation.

We develop a conservative, second order accurate fully implicit discretization of ternary (three-phase) Cahn-Hilliard (CH) systems that has an associated discrete energy functional. This is an extension of our work for two-phase systems. We analyze and prove convergence of the scheme. To efficiently solve the discrete system at the implicit time-level, we use a nonlinear multigrid method. The resulting scheme is efficient, robust and there is at most a 1st order time step constraint for stability. We demonstrate convergence of our scheme numerically and we present several simulations of phase transitions in ternary systems.

In this paper, we propose a class of exact artificial boundary conditions for the numerical solution of the Schrödinger equation on unbounded domains in two-dimensional cases. After we introduce a circular artificial boundary, we get an initial-boundary problem on a disc enclosed by the artificial boundary which is equivalent to the original problem. Based on the Fourier series expansion and the special functions techniques, we get the exact artificial boundary condition and a series of approximating artificial boundary conditions. When the potential function is independent of the radiant angle θ, the problem can be reduced to a series of one-dimensional problems. That can reduce the computational complexity greatly. Our numerical examples show that our method gives quite good numerical solutions with no numerical reflections.

This paper, which is a sequel to Benedetto-Caglioti-Golse-Pulvirenti, Comput. Math. Appl. 38 (1999), p. 121-131, considers as a starting point a mean-field equation for the dynamics of a gas of particles interacting via dissipative binary collisions. More precisely, we are concerned with the case where these particles are immersed in a thermal bath modeled by a linear Fokker-Planck operator. Two different scalings are considered for the resulting equation. One concerns the case of a thermal bath at finite temperature and leads formally to a nonlinear diffusion equation. The other concerns the case of a thermal bath at infinite temperature and leads formally to an isentropic Navier-Stokes system. Both formal limits rest on the mathematical properties of the linearized mean-field operator which are established rigorously, and on a Hilbert or Chapman-Enskog expansion.

We present a multiscale method for a class of problems that are locally self-similar in scales and hence do not have scale separation. Our method is based on the framework of the heterogeneous multiscale method (HMM). At each point where macroscale data is needed, we perform several small scale simulations using the microscale model, then using the results and local selfsimilarity to predict the needed data at the scale of interest. We illustrate this idea by computing the effective macroscale transport of a percolation network at the percolation threshold.