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Stochastic Algorithms for Solving the Dirichlet Boundary Value Problem for Certain Second-Order Elliptic Equations with Discontinuous Coefficients

Abstract

Stochastic algorithms for solving the Dirichlet boundary value problem for a second-order elliptic equation with coefficients having a discontinuity on a smooth surface are considered. It is assumed that the solution is continuous and its normal derivatives on the opposite sides of the discontinuity surface are consistent. A mean value formula in a ball (or an ellipsoid) is proposed and proved. This formula defines a random walk in the domain and provides statistical estimators (on its trajectories) for finding a Monte Carlo solution of the boundary value problem at the initial point of the walk.

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ACKNOWLEDGMENTS

The authors are grateful to the reviewer for comments that have helped improve the manuscript.

Funding

This work was supported by the Russian Science Foundation, grant no. 19-11-00020.

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Correspondence to A. N. Kuznetsov or A. S. Sipin.

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The authors declare that they have no conflicts of interest.

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Translated by I. Ruzanova

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Kuznetsov, A.N., Sipin, A.S. Stochastic Algorithms for Solving the Dirichlet Boundary Value Problem for Certain Second-Order Elliptic Equations with Discontinuous Coefficients. Comput. Math. and Math. Phys. 62, 248–253 (2022). https://doi.org/10.1134/S0965542522020099

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  • DOI: https://doi.org/10.1134/S0965542522020099

Keywords:

  • elliptic operator
  • boundary value problem
  • mean value formula
  • random walk
  • stochastic algorithm
  • unbiased estimator