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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The authors are grateful to the reviewer for comments that have helped improve the manuscript.
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This work was supported by the Russian Science Foundation, grant no. 19-11-00020.
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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