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

Abstract

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 [41], our two-level stochastic collocation method utilizes a twogrid 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.

Second-order efficient algorithm for coupled nonlinear model of groundwater transport system

“…With this condition, it will spend relatively much time to solve the nonlinear problem on the coarse grid. In [34] and [35], Chen and Zheng et al generalized the two-grid technique for solving nonlinear problems in two different domains. This idea minimized the computational cost.…”

Two grid finite element discretization method for semi‐linear hyperbolic integro‐differential equations

“…The main idea of this method is using a coarse-grid space to produce a rough approximation of the solution for nonlinear problems, and then use it as the initial guess for one Newton-like iteration on the fine grid. Two-grid discretization method has been widely used for different kinds of problems, such as elliptic Equations [9,10], parabolic equations [12][13][14][15][16][17], eigenvalue problems [18][19][20] stochastic partial differential equations [21] and fractional differential equations [22,23]. The two-grid discretization idea is also used for nonlinear coupled equations, such as the complicated miscible displacement problems [24][25][26] and fluid flow in porous media [27].…”

“…The two‐grid discretization idea is also used for nonlinear coupled equations, such as the complicated miscible displacement problems and fluid flow in porous media . The technique is also combined with different kinds of existed numerical methods for discretizing continuous partial differential equations, for instance, two‐grid finite difference method , two‐grid finite element method , two‐grid finite volume method and two‐grid spectral collocation method . The two‐grid idea was first proposed for nonlinear integral equations in and it has been further developed in the studies of .…”