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

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 TH with a low level stochastic collocation (corresponding to the polynomial space PP) and solve linearized equations on a fine mesh Th using high level stochastic collocation (corresponding to the polynomial space Pp). 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 Th and Pp. 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.