On a Fictitious Domain Method for Unilateral Problems

This contribution deals with numerical realization of elliptic boundary value problems with unilateral boundary conditions using a fictitious domain method. Any fictitious domain formulation [2] extends the original problem defined in a domain ω to a new (fictitious) domainΩ with a simple geometry (e.g. a box) which contains ω . The main advantage consists in possibility to use a uniform mesh in Ω leading to a structured stiffness matrix. This enables us to apply highly efficient multiplying procedures [6]. Fictitious domain formulations of problems with the classical Dirichlet or Neumann boundary conditions lead after a finite element discretization typically to algebraic saddle-point systems. For their solution one can use the algorithm studied in [4] that combines the Schur complement reduction with the null-space method. The situation is not so easy for unilateral problems since their weak formulation contains a non-differentiable projection operator. Fortunately, a resulting algebraic representation is described by a system that is semi-smooth in the sense of [1] so that a generalized Newton method can be applied. This method has been already used in [5] for solving complementarity problems. In our case each Newton step relates to a mixed Dirichlet-Neumann problem and therefore the algorithm from [4] can be used for solving inner linear systems. Due to the superlinear convergence rate of the Newton iteration [1], the computations are only slightly more expensive than the solution of pure Dirichlet or Neumann problems. In this paper we compare two variants of the fictitious domain method. The first one enforces unilateral conditions by Langrangemultipliers defined on the boundary γ of the original domainω . Therefore the fictitious domain solution has a singularity