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
Δ2*u* + *au* = *f*
in the piecewise bicubic Hermit interpolations subspace of *W*22(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.

Abstract

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33-43