# American Institute of Mathematical Sciences

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October  2018, 11(5): 1139-1156. doi: 10.3934/krm.2018044

## Uniform spectral convergence of the stochastic Galerkin method for the linear semiconductor Boltzmann equation with random inputs and diffusive scaling

Received  June 2017 Revised  October 2017 Published  May 2018

Fund Project: The author was supported in part by Prof. Shi Jin’s NSF grants DMS-1522184 and DMS-1107291: RNMS KI-Net.

In this paper, we study the generalized polynomial chaos (gPC) based stochastic Galerkin method for the linear semiconductor Boltzmann equation under diffusive scaling and with random inputs from an anisotropic collision kernel and the random initial condition. While the numerical scheme and the proof of uniform-in-Knudsen-number regularity of the distribution function in the random space has been introduced in [15], the main goal of this paper is to first obtain a sharper estimate on the regularity of the solution-an exponential decay towards its local equilibrium, which then lead to the uniform spectral convergence of the stochastic Galerkin method for the problem under study.

Citation: Liu Liu. Uniform spectral convergence of the stochastic Galerkin method for the linear semiconductor Boltzmann equation with random inputs and diffusive scaling. Kinetic & Related Models, 2018, 11 (5) : 1139-1156. doi: 10.3934/krm.2018044
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