A two-level stochastic collocation method for semilinear elliptic equations with random coefficients

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS

卷:315

Article

Chen, Luoping¹  Zheng, Bin²  Lin, Guang³  Voulgarakis, Nikolaos⁴ 

[1] Southwest Jiaotong Univ, Sch Math, Chengdu 611756, Peoples R China

[2] Pacific Northwest Natl Lab, Adv Comp Math & Data Div, Richland, WA 99352 USA

[3] Purdue Univ, Sch Mech Engn, Dept Math, W Lafayette, IN 47907 USA

[4] Washington State Univ, Dept Math, Richland, WA 99354 USA

关键词: Semilinear problems;    Random coefficients;    Two-grid;    Finite element;    Stochastic collocation;   

DOI  :  10.1016/j.cam.2016.10.030

来源: Elsevier

【 摘 要 】

In this work, we propose a novel two-level discretization for solving semilinear elliptic equations with random coefficients. Motivated by the two-grid method for deterministic partial differential equations (PDEs) introduced by Xu (1994), our two-level stochastic collocation method utilizes a two-grid finite element discretization in the physical space and a two-level collocation method in the random domain. In particular, we solve semilinear equations on a coarse mesh T-H with a low level stochastic collocation (corresponding to the polynomial space P-p) and solve linearized equations on a fine mesh T-h using high level stochastic collocation (corresponding to the polynomial space P-p). We prove that the approximated solution obtained from this method achieves the same order of accuracy as that from solving the original semilinear problem directly by stochastic collocation method with T-h and P-p. The two-level method is computationally more efficient than the standard stochastic collocation method for solving nonlinear problems with random coefficients. Numerical experiments are provided to verify the theoretical results. (C) 2016 Elsevier B.V. All rights reserved.

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