16. Finite Difference Method with Fictitious Domain Applied to a Dirichlet Problem

Here f ∈ L(Ω) , g ∈ H(∂Ω) and Ω is a bounded domain in R with the smooth boundary ∂Ω ( see Figure 1 ). The method of lines for solving Problem I works well if Ω is a rectangular domain since the finite difference solution is expressed explicitly by use of eigenvalues and eigenvectors for the finite difference scheme([BGN70], [Nak65]). But one says that this method seems difficult to be applied to the case where Ω is not a rectangular domain. However the solution algorithm using the fictitious domain and the domain decomposition has been developed recently ( [AKP95], [GPP94], [HH99], [FKK95], [KK99], [MKM86]). Hence from this point of view we shall propose a numerical algorithm by the method of lines coupled with a fictitious domain in this paper. First of all, we embed Ω in a rectangular domain Π whose boundary ∂Π consists of straight lines parallel to axes and set Ω1 = Π \ (Ω ∪ ∂Ω) ( see Figure 2 ). Then Π is called a fictitious domain.