A D.C. Algorithm via Convex Analysis Approach for Solving a Location Problem Involving Sets

We study a location problem that involves a weighted sum of distances to closed convex sets. As several of the weights might be negative, traditional solution methods of convex optimization are not applicable. After obtaining some existence theorems, we introduce a simple, but effective, algorithm for solving the problem. Our method is based on the Pham Dinh - Le Thi algorithm for d.c. programming and a generalized version of the Weiszfeld algorithm, which works well for convex location problems

Exact internal waves of a Boussinesq system

We consider a Boussinesq system describing one-dimensional internal waves which develop at the boundary between two immiscible fluids, and we restrict to its traveling waves. The method which yields explicitly all the elliptic or degenerate elliptic solutions of a given nonlinear, any order algebraic ordinary differential equation is briefly recalled. We then apply it to the fluid system and, restricting in this preliminary report to the generic situation, we obtain all the solutions in that class, including several new solutions.Comment: 11 pages, Waves and stability in continuous media, Palermo, 28 June-1 July 2009. Eds. A.Greco, S.Rionero and T.Ruggeri (World scientific, Singapore, 2010

Analysis and discretization of the volume penalized Laplace operator with Neumann boundary conditions

We study the properties of an approximation of the Laplace operator with Neumann boundary conditions using volume penalization. For the one-dimensional Poisson equation we compute explicitly the exact solution of the penalized equation and quantify the penalization error. Numerical simulations using finite differences allow then to assess the discretisation and penalization errors. The eigenvalue problem of the penalized Laplace operator with Neumann boundary conditions is also studied. As examples in two space dimensions, we consider a Poisson equation with Neumann boundary conditions in rectangular and circular domains

Approximation of the Laplace and Stokes operators with Dirichlet boundary conditions through volume penalization: a spectral viewpoint

We report the results of a detailed study of the spectral properties of Laplace and Stokes operators, modified with a volume penalization term designed to approximate Dirichlet conditions in the limit when a penalization parameter, $\eta$, tends to zero. The eigenvalues and eigenfunctions are determined either analytically or numerically as functions of $\eta$, both in the continuous case and after applying Fourier or finite difference discretization schemes. For fixed $\eta$, we find that only the part of the spectrum corresponding to eigenvalues $\lambda \lesssim \eta^{-1}$ approaches Dirichlet boundary conditions, while the remainder of the spectrum is made of uncontrolled, spurious wall modes. The penalization error for the controlled eigenfunctions is estimated as a function of $\eta$ and $\lambda$. Surprisingly, in the Stokes case, we show that the eigenfunctions approximately satisfy, with a precision $O(\eta)$, Navier slip boundary conditions with slip length equal to $\sqrt{\eta}$. Moreover, for a given discretization, we show that there exists a value of $\eta$, corresponding to a balance between penalization and discretization errors, below which no further gain in precision is achieved. These results shed light on the behavior of volume penalization schemes when solving the Navier-Stokes equations, outline the limitations of the method, and give indications on how to choose the penalization parameter in practical cases