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

  • Numerical Methods for Solution of Equations
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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 .

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References

  1. S. A. Voitsekhovskii, "Approximate solution of the Dirichlet problem for second-order elliptical equation in domains of arbitrary shape," Vychisl. Prikl. Mat., No. 54, 26–31 (1984).

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Authors
  1. S. A. Voitsekhovskii
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  2. V. N. Novichenko
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Additional information

Kiev University. Translated from Vychislitel'naya i Prikladnaya Matematika, No. 75, pp. 20–25, 1991.

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Voitsekhovskii, S.A., Novichenko, V.N. Convergence bounds of difference schemes for secondorder elliptical equations in domains of arbitrary shape. J Math Sci 72, 3061–3065 (1994). https://doi.org/10.1007/BF01259471

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  • DOI: https://doi.org/10.1007/BF01259471

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