Abstract
In this paper, we present numerical methods to implement the probabilistic representation of third kind (Robin) boundary problem for the Laplace equations. The solution is based on a Feynman–Kac formula for the Robin problem which employs the standard reflecting Brownian motion (SRBM) and its boundary local time arising from the Skorokhod problem. By simulating SRBM paths through Brownian motion using Walk on Spheres method, approximation of the boundary local time is obtained and the Feynman–Kac formula is then calculated by evaluating the average of all path integrals over the boundary under a measure defined through the local time. Numerical results demonstrate the accuracy and efficiency of the proposed method for finding a local solution of the Laplace equations with Robin boundary conditions.
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Audus, D.J., Hassan, A.M., Garboczi, E.J., Douglas, J.F.: Interplay of particle shape and suspension properties: a study of cube-like particles. Soft Matter 11(17), 3360–3366 (2015)
Binder, I., Braverman, M.: The rate of convergence of the walk of sphere algorithm. Geom. Funct. Anal. 22, 558–587 (2012)
Burdzy, K., Chen, Z., Sylvester, J.: The heat equation and reflected Brownian motion in time-dependent domains. Annu. Probab. 32(1B), 775–804 (2004)
Chung, K.L.: Green, Brown, and Probability and Brownian Motion on the Line. World Scientific Pub Co Inc, Singapore (2002)
Douglas, J.F.: Integral equation approach to condensed matter relaxation. J. Phys. Condens. Matter 11(10A), A329 (1999)
Feynman, R.P.: Space-time approach to nonrelativistic quantum mechanics. Rev. Mod. Phys. 20, 367–387 (1948)
Freidlin, M.: Functional Integration and Partial Differential Equations. Princeton University Press, Princeton (1985)
Hsu, (Elton) P.: Reflecting Brownian motion, boundary local time and the Neumann problem, Dissertation Abstracts International Part B: Science and Engineering [DISS. ABST. INT. PT. B- SCI. ENG.], Vol. 45, No. 6 (1984)
Hwang, C.O., Mascagni, M., Given, J.A.: A Feynman–Kac path-integral implementation for Poisson’s equation using an h-conditioned Green’s function. Math. Comput. Simul. 62(3), 347–355 (2003)
Kac, M.: On distributions of certain Wiener functionals. Trans. Am. Math. Soc. 65, 1–13 (1949)
Kac, M.: On some connections between probability theory and differential and integral equations. In: Proceedings of 2nd Berkeley Symposium Math. Stat. and Probability, vol. 65, pp. 189–215 (1951)
Karatzas, I., Shreve, S.E.: Brownian Motion and Stochastic Calculus. Springer, New York (1988)
Lejay, A., Maire, S.: New Monte Carlo schemes for simulating diffusions in discontinuous media. J. Comput. Appl. Math. 245, 97–116 (2013)
Lions, P.L., Sznitman, A.S.: Stochastic differential equations with reflecting boundary conditions. Commun. Pure Appl. Math. 37(4), 511–537 (1984)
Maire, S., Tanré, E.: Monte Carlo approximations of the Neumann problem. Monte Carlo Methods Appl. 19(3), 201–236 (2013)
Morillon, J.-P.: Numerical solutions of linear mixed boundary value problems using stochastic representations. Int. J. Numer. Methods Eng. 40, 387–405 (1997)
Müller, M.E.: Some continuous Monte Carlo methods for the Dirichlet problem. Ann. Math. Stat. 27(3), 569–589 (1956)
Øksendal, B.: Stochastic Differential Equations. Springer, Berlin (2003)
Papanicolaou, V.G.: The probabilistic solution of the third boundary value problem for second order elliptic equations. Probab. Theory Relat. Fields 87, 27–77 (1990)
Sabelfeld, K.K., Simonov, N.A.: Random walks on boundary for solving PDEs, Walter de Gruyter (1994)
Skorokhod, A.V.: Stochastic equations for diffusion processes in a bounded region. Theory Probab. Appl. 6(3), 264–274 (1961)
Souza de Cursi, J.E.: Numerical methods for linear boundary value problems based on Feynman–Kac representations. Math. Comput. Simul. 36(1), 1–16 (1994)
Tanaka, H.: Stochastic differential equations with reflecting boundary condition in convex regions. Hiroshima Math. J. 9(1), 163–177 (1979)
Yan, C., Cai, W., Zeng, X.: A parallel method for solving Laplace equations with Dirichlet data using local boundary integral equations and random walks. SIAM J. Sci. Comput. 35(4), B868–B889 (2013)
Zhou, Y., Cai, W., Hsu, (Elton) P.: Local Time of Reflecting Brownian Motion and Probabilistic Representation of the Neumann Problem, Preprint (2015)
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The authors Y.J.Z. and W.C. acknowledge the support of the National Science Foundation (DMS-1315128) and the National Natural Science Foundation of China (No. 91330110) for the work in this paper.
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Zhou, Y., Cai, W. Numerical Solution of the Robin Problem of Laplace Equations with a Feynman–Kac Formula and Reflecting Brownian Motions. J Sci Comput 69, 107–121 (2016). https://doi.org/10.1007/s10915-016-0184-y
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DOI: https://doi.org/10.1007/s10915-016-0184-y