An Adaptive Fictitious Domain Method for elliptic problems

In the Fictitious Domain Method with Lagrange multiplier (FDM) the physical domain is embedded into a simpler but larger domain called the fictitious domain. The partial differential equation is extended to the fictitious domain using a Lagrange multiplier to enforce the prescribed boundary conditions on the physical domain while all the other data are extended to the fictitious domain. This lead to a saddle point system coupling the Lagrange multiplier and the extended solution of the original problem. At the discrete level, the Lagrange multiplier is approximated on subdivisions of the physical boundary while the extended solution is approximated on partitions of the fictitious domain. A significant advantage of the FDM is that no conformity between these two meshes is required. However, a restrictive compatibility conditions between the mesh-sizes must be enforced to ensure that the discrete saddle point system is well-posed. In this paper, we present an adaptive fictitious domain method (AFDM) for the solution of elliptic problems in two dimensions. The method hinges upon two modules ELLIPTIC and ENRICH which iteratively increase the resolutions of the approximation of the extended solution and the multiplier, respectively. The adaptive algorithm AFDM is convergent without any compatibility condition between the two discrete spaces. We provide numerical experiments illustrating the performances of the proposed algorithm.