A FICTITIOUS DOMAIN METHOD FOR DIRICHLET PROBLEM AND APPLICATIONS

In this article we discuss the solution of the Dirichlet problem for a class of elliptic operators by a Lagrange multiplier/fictitious domain method. This approach allows the use of regular grids and therefore of fast specialized solvers for problems on complicated geometries; the resulting saddle-point system can be solved by an Uzawa/conjugate gradient algorithm. In the case of two-dimensional problems, a quasi-optimal preconditioner has been found by Fourier analysis and numerical experiments confirm its nice scaling properties. The resulting methodology is applied to a nonlinear time dependent problem, namely the flow of a viscous-plastic medium in a cylindrical pipe showing the potential of this methodology for some classes of nonlinear problems.