It is well-known that the “standard” oblique derivative problem, $\Delta u=0$ in $\Omega$, $\partial u/\partial \nu -u=0$ on $\partial \Omega$ ($\nu$ is the unit inner normal) has a unique solution even when the boundary condition is not assumed to hold on the entire boundary. When the boundary condition is modified to satisfy an obliqueness condition, the behavior at a single boundary point can change the uniqueness result. We give two simple examples to demonstrate what can happen.

In this paper, we present numerical methods for the determination of solitons, that consist in spatially localized stationary states of nonlinear scalar equations or coupled systems arising in nonlinear optics. We first use the well-known shooting method in order to find excited states (characterized by the number $k$ of nodes) for the classical nonlinear Schrödinger equation. Asymptotics can then be derived in the limits of either large $k$ are large nonlinear exponents $\sigma$. In a second part, we compute...

In this paper, we present numerical methods for the determination of solitons, that consist in spatially localized stationary states of nonlinear scalar equations or coupled systems arising in nonlinear optics. We first use the well-known shooting method in order to find excited states (characterized by the number k of nodes) for the classical nonlinear Schrödinger equation. Asymptotics can then be derived in the limits of either large k are large nonlinear exponents σ. In a second part, we compute...

These notes give a state of the art of numerical homogenization methods for linear elliptic equations. The guideline of these notes is analysis. Most of the numerical homogenization methods can be seen as (more or less different) discretizations of the same family of continuous approximate problems, which H-converges to the homogenized problem. Likewise numerical correctors may also be interpreted as approximations of Tartar’s correctors. Hence the...

The paper deals with numerical solution of internal flow problems. It mentions a long tradition of mathematical modeling of internal flow, especially transonic flow at our department. Several models of flow based on potential equation, Euler equations, Navier-Stokes and Reynolds averaged Navier-Stokes equations with proper closure are considered. Some mathematical and numerical properties of the model are mentioned and numerical results achieved by in-house developed methods are presented.

In [Progress Math.233 (2005)], David suggested the existence of a new type of global minimizers for the Mumford-Shah functional in ${𝐑}^3$. The singular set of such a new minimizer belongs to a three parameters family of sets $(0<{\delta }_1,{\delta }_2,{\delta }_3<\pi )$. We first derive necessary conditions satisfied by global minimizers of this family. Then we are led to study the first eigenvectors of the Laplace-Beltrami operator with Neumann boundary conditions on subdomains of ${𝐒}^2$ with three reentrant corners. The necessary conditions are...

Assuming an incident wave to be a field source, we calculate the field potential in a neighborhood of an inhomogeneous body. This problem which has been formulated in ${𝐑}^3$can be reduced to a bounded domain. Namely, a boundary condition for the potential is formulated on a sphere. Then the potential satisfies a well posed boundary value problem in a ball containing the body. A numerical approximation is suggested and its convergence is analyzed.

This contribution gives an overview of current research in applying object oriented programming to scientific computing at the computational mechanics laboratory (LABMEC) at the school of civil engineering – UNICAMP. The main goal of applying object oriented programming to scientific computing is to implement increasingly complex algorithms in a structured manner and to hide the complexity behind a simple user interface. The following areas are current topics of research and documented within the...

This contribution gives an overview of current research in applying object oriented programming to scientific computing at the computational mechanics laboratory (LABMEC) at the school of civil engineering – UNICAMP. The main goal of applying object oriented programming to scientific computing is to implement increasingly complex algorithms in a structured manner and to hide the complexity behind a simple user interface. The following areas are current topics of research and documented within the...

A priori estimates and strong solvability results in Sobolev space $W^{2,p}\left(\Omega \right)$, $1<p<\infty$ are proved for the regular oblique derivative problem $$\left\{\begin{array}{c}\sum _{i,j=1}^n{a}^{ij}\left(x\right)\frac{{\partial }^{2}u}{\partial x_i\partial x_j}=f\left(x\right)\text{a.e.}\Omega \hfill \\ \frac{\partial u}{\partial \ell }+\sigma \left(x\right)u=\varphi \left(x\right)\text{on}\partial \Omega \hfill \end{array}\right.$$ when the principal coefficients $a^{ij}$ are $VMO\cap L^{\infty }$ functions.