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

作者：Chen, Luoping; Zheng, Bin^*; Lin, Guang; Voulgarakis, Nikolaos

来源：Journal of Computational and Applied Mathematics, 2017, 315: 195-207.

DOI：10.1016/j.cam.2016.10.030

• Semilinear problems
• Random coefficients
• Two-grid
• Finite element
• Stochastic collocation

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.

• 出版日期2017-5-1
• 单位西南交通大学