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On Probabilistic Analytical and Numerical Approaches for Divergence Form Operators with Discontinuous Coefficients

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Advances in Numerical Simulation in Physics and Engineering

Part of the book series: SEMA SIMAI Springer Series ((SEMA SIMAI,volume 3))

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Abstract

In this paper we review some recent results on stochastic analytical and numerical approaches to parabolic and elliptic partial differential equations involving a divergence form operator with a discontinuous coefficient and a compatibility transmission condition.

In the one-dimensional case existence and uniqueness results for such PDEs can be obtained by stochastic methods. The probabilistic interpretation of the solutions allows one to develop and analyze a low complexity Monte Carlo numerical resolution method. In addition, it allows to get accurate pointwise estimates for the derivatives of the solutions from which sharp convergence rate estimates are deduced for the stochastic numerical method.

A stochastic approach is also developed for the linearized Poisson–Boltzmann equation in Molecular Dynamics. As in the one-dimensional case, the probabilistic interpretation of the solution involves the solution of a SDE including a non standard local time term related to the discontinuity interface. We present an extended Feynman–Kac formula for the Poisson–Boltzmann equation. This formula justifies various probabilistic numerical methods to approximate the free energy of a molecule and bases error analyzes.

We finally present probabilistic interpretations of the non-linearized Poisson–Boltzmann equation in terms of backward stochastic differential equations.

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Authors and Affiliations

  1. 2002 Route des Lucioles, BP 93, 06901, Sophia Antipolis Cedex, France

    Denis Talay

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  1. Denis Talay
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Correspondence to Denis Talay .

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Editors and Affiliations

  1. Facultad de Ciencias, Universidad de Málaga, Málaga, Málaga, Spain

    Carlos Parés

  2. Facultad de Informática, Universidade da Coruña, A Coruña, La Coruña, Spain

    Carlos Vázquez

  3. Centre de Mathématiques Appliquées, CNRS and Ecole Polytechnique, Paris, Paris, France

    Frédéric Coquel

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Talay, D. (2014). On Probabilistic Analytical and Numerical Approaches for Divergence Form Operators with Discontinuous Coefficients. In: Parés, C., Vázquez, C., Coquel, F. (eds) Advances in Numerical Simulation in Physics and Engineering. SEMA SIMAI Springer Series, vol 3. Springer, Cham. https://doi.org/10.1007/978-3-319-02839-2_7

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