A smoothness preserving fictitious domain method for elliptic boundary-value problems

Abstract

We introduce a new fictitious domain method for the solution of second-order elliptic boundary-value problems with Dirichlet or Neumann boundary conditions on domains with C2 boundary. The main advantage of this method is that it extends the solutions smoothly, which leads to better performance by achieving higher accuracy with fewer degrees of freedom. The method is based on a least-squares interpretation of the fundamental requirements that the solution produced by a fictitious domain method should satisfy. Careful choice of discretization techniques, together with a special solution strategy, leads then to smooth solutions of the resulting underdetermined problem. Numerical experiments are provided which illustrate the performance and flexibility of the approach.