Convergence bounds of difference schemes for secondorder elliptical equations in domains of arbitrary shape

Abstract

The fictitious domain method is applied to construct a difference scheme for the first boundary-value problem for elliptical equations of second order in domains of arbitrary shape. The rate of convergence bound $$\parallel \hat y - \bar u\parallel _{W_2^1 (\Omega _0 )} \leqslant Mh^{\frac{1}{3}} \parallel f\parallel _{L_2 (\Omega )} ,$$ is proved, where ŷ is the polylinear extension of the solution of the difference problem, ū is the solution of the original problem continued as zero in Ω 1 ,and Ω 1 is the complement of the domain Ωto the rectangle Ω 0 .