Rate of convergence bounds of difference schemes for fourth-order elliptical equations in regions of arbitrary shape

Abstract

The fictitious domain method is applied to construct a difference scheme for the first boundary-value problem for fourth-order elliptical equations in regions of arbitrary shape. The rate of convergence bound

$$\left\| {y - \bar u} \right\|_{W_2^k (\omega )} \leqslant Mh^{\tfrac{{3 - k}}{2}} \left\| f \right\|_{L_2 (\Omega )} ,k = 1,2,$$

is proved. Here y is the solution of the difference problem, — u is the solution of the original problem continued as zero Ω₁, where Ω₁ is the complement of the region Ω to the rectangle Ω₀.