A fictitious domain approach with spread interface for elliptic problems with general boundary conditions

摘要

The aim of this article is to solve second-order elliptic problems in an original physical domain using a fictitious domain method with a spread interface approach. The main idea of the fictitious domain approach consists in immersing the original domain of study into a geometrically bigger and simpler one called fictitious domain. As the spatial discretization is being performed in the fictitious domain, this method allows the use of structured meshes. The discretization is not boundary-fitted to the original physical domain. This paper describes several ways to impose Dirichlet, Robin or Neumann boundary conditions on a spread immersed interface, without locally modifying the numerical scheme and without using Lagrange multipliers. The numerical applications focus on diffusion and convection problems in the unit disk, with Dirichlet or Robin boundary conditions. For such problems, analytical solutions can be determined for a correctly chosen source term. The numerical resolution is performed using a L_1 Finite Element scheme. The spread interface approach is then combined with a local adaptive mesh refinement algorithm in order to increase the precision in the vicinity of the immersed boundary. The L~2 norm of the errors is computed in order to evaluate the capability of the method. Immersed boundaries are found in many industrial applications like two-phase flow simulations, fluid/structure interaction, etc. This article represents a first step towards the simulation of these kinds of applications.