Boundary Value Problems in Some Ramified Domains with a Fractal Boundary: Analysis and Numerical Methods. Part II: Non homogeneous Neumann Problems.

摘要

This paper is devoted to numerical methods for solving Poisson problems in self-similar ramified domains of $R^2$ with a fractal boundary. It is proved that a sequence of solutions to some nonhomogeneous Neumann problems posed on domains obtained by interrupting the fractal construction after a finite number of generations, converges to the solution of a Neumann problem posed in the whole domain. To define the Neumann problem on the infinitely ramified domain and for proving the above mentioned convergence, extension and trace results are given. Then, a method for computing the solution is proposed an analyzed. In particular, it is shown that the small scales of the Neumann data are damped exponentially fast away from the boundary. A self similar finite element method is developed and tested.