A general fictitious domain method with immersed jumps and multilevel nested structured meshes

Abstract

This study addresses a new fictitious domain method for elliptic problems in order to handle general and possibly mixed embedded boundary conditions (E.B.C.): Robin, Neumann and Dirichlet conditions on an immersed interface. The main interest of this fictitious domain method is to use simple structured meshes, possibly uniform Cartesian nested grids, which do not generally fit the interface but define an approximate one. A cell-centered finite volume scheme with a non-conforming structured mesh is derived to solve the set of equations with additional algebraic transmission conditions linking both flux and solution jumps through the immersed approximate interface. Hence, a local correction is devised to take account of the relative surface ratios in each control volume for the Robin or Neumann boundary condition. Then, the numerical scheme conserves the first-order accuracy with respect to the mesh step. This opens the way to combine the E.B.C. method with a multilevel mesh refinement solver to increase the precision in the vicinity of the interface. Such a fictitious domain method is very efficient: the L 2- and L ∞-norm errors vary like O ( h l ∗ ) where h l ∗ is the grid step of the finest refinement level around the interface until the residual first-order discretization error of the non-refined zone is reached. The numerical results reported here for convection–diffusion problems with Dirichlet, Robin and mixed (Dirichlet and Robin) boundary conditions confirm the expected accuracy as well as the performances of the present method.