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A Multi-modes Monte Carlo Interior Penalty Discontinuous Galerkin Method for the Time-Harmonic Maxwell’s Equations with Random Coefficients

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Abstract

This paper develops an efficient Monte Carlo interior penalty discontinuous Galerkin (IP-DG) method for electromagnetic wave propagation in random media. This method is based on a multi-modes expansion of the solution to the time-harmonic random Maxwell equations. It is shown that each mode function satisfies a Maxwell system with random sources defined recursively. An unconditionally stable IP-DG method is employed to discretize the nearly deterministic Maxwell system and the Monte Carlo method combined with an efficient acceleration strategy is proposed for computing the mode functions and the statistics of the electromagnetic wave. A complete error analysis is established for the proposed multi-modes Monte Carlo IP-DG method. It is proved that the proposed method converges with an optimal order for each of three levels of approximations. Numerical experiments are provided to validate the theoretical results and to gauge the performance of the proposed numerical method and approach.

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Authors and Affiliations

  1. Department of Mathematics, The University of Tennessee, Knoxville, TN, 37996, USA

    Xiaobing Feng

  2. Department of Mathematics and Statistics, Auburn University, Auburn, AL, 36849, USA

    Junshan Lin

  3. Department of Mathematics and Statistics, The University of West Florida, Pensacola, FL, 32514, USA

    Cody Lorton

Authors
  1. Xiaobing Feng
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  2. Junshan Lin
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  3. Cody Lorton
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Correspondence to Cody Lorton.

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This work was partially supported by the NSF Grants DMS-1620168 and DMS-1719851.

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Feng, X., Lin, J. & Lorton, C. A Multi-modes Monte Carlo Interior Penalty Discontinuous Galerkin Method for the Time-Harmonic Maxwell’s Equations with Random Coefficients. J Sci Comput 80, 1498–1528 (2019). https://doi.org/10.1007/s10915-019-00986-3

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  • DOI: https://doi.org/10.1007/s10915-019-00986-3

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