Fictitious domain method for fourth-order elliptic problem

In this paper, we study the convergence of the fictitious domain method for solving a system of grid equations for the finite element method that approximates the third boundary problem for the differential equation Δ2u + au = f in the piecewise bicubic Hermit interpolations subspace of W22(Ω) on a rectangular grid. The main operation on each step of the method is a double inversion of the FEM operator for the trivial boundary value Dirichlet problem for the Poisson equation in a rectangle, which interior includes the rectangle composed domain Ω. For this purpose an inner iterational process is built. The speed of the convergence of two-level iterative method does not depend on the grid parameter, so this method gives the solution with O(h-2(ln h-1)2 ln ε-1) arithmetic operations, where ε is an accuracy of the solution.