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A two-level stochastic collocation method for semilinear elliptic equations with random coefficients
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 \cite{xu1994novel}, 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 with a low level
stochastic collocation (corresponding to the polynomial space
) and solve linearized equations on a fine mesh
using high level stochastic collocation (corresponding to the
polynomial space ). 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 and
. 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.Comment: 20 pages, 2 figure