The numerical solution of the free boundary Bernoulli problem is addressed. An iterative method based on a level-set formulation and boundary element method is proposed. Issues related to the implementation, the accuracy and the generality of the method are discussed. The efficiency of the approach is illustrated by numerical results.

We report on recent progress in the study of evolution processes involving degenerate parabolic equations which may exhibit free boundaries. The equations we have selected follow two recent trends in diffusion theory: considering anomalous diffusion with long-range effects, which leads to fractional operators or other operators involving kernels with large tails; and the combination of diffusio...

This contribution combines a shape optimization approach to free boundary value problems of Bernoulli type with an embedding domain technique. A theoretical framework is developed which allows to prove continuous dependence of the primal and dual variables in the resulting saddle point problems with respect to the domain. This ensures the existence of a solution of a related shape optimization ...

We provide a free discontinuity approach to a class of shape optimization problems involving Robin conditions on the free boundary. More precisely, we identify a large family of domains on which such problems are well posed in a way that the extended problem can be considered a relaxed version of the corresponding one on regular domains, we prove existence of a solution and obtain some qualitat...

The present paper is dedicated to the numerical solution of stationary free boundary problem by means of shape optimization. We assume that the state satisfies the Dirichlet problem and track the Neumann data at the free boundary. The gradient and Hessian of the shape functional under consideration are computed. By analysing the shape Hessian in case of matching data a sufficient criterion for ...

singular perturbation problems have been studied by many mathematicians. since the approximate solutions of these problems are as the sum of internal solution (boundary layer area) and external ones, the formation or non-formation of boundary layers should be specified. this paper, investigates this issue for a singular perturbation problem including a first order differential equation with gen...

We derive Newton-type solution algorithms for a Bernoulli-type freeboundary problem at the continuous level. The Newton schemes are obtained by applying Hadamard shape derivatives to a suitable weak formulation of the freeboundary problem. At each Newton iteration, an updated free boundary position is obtained by solving a boundary-value problem at the current approximate domain. Since the boun...

We consider the existence of multi-peak solutions to two types of free bound­ ary problems arising in confined plasma and steady vortex pair under conditions on the nonlinearity we believe to be almost optimal. Our results show that the “core” of the solution has multiple connected components, whose boundary called free boundary of the problems consists approximately of spheres which shrink to ...

The Incompressible Smoothed Particle Hydrodynamic (ISPH) is one of the particle methods and commonly used to solve some complicated physical problems including free surface flow problems. The study regarding the boundary treatment has become an active research area in the mesh-free or particle method recently for measuring the accurate and robust pressure near the boundary. The penetrations of ...