A Two-Level Sparse Grid Collocation Method for Semilinear Stochastic Elliptic Equation

In this work, we investigate a novel two-level discretization method for the elliptic equations with random input data. Motivated by the two-grid method for deterministic nonlinear partial differential equations introduced by Xu [36], our two-level discretization method uses a two-grid finite element method in the physical space and a two-scale stochastic collocation method with sparse grid in the random domain. Specifically, we solve a semilinear equations on a coarse mesh T-H(D) with small scale of sparse collocation points eta(L, N) and solve a linearized equations on a fine mesh T h(D) using large scale of sparse collocation points eta(l, N) (where eta(L, N), eta(l, N) are the numbers of sparse grid with respect to different levels L, l in N dimensions). Moreover, an error correction on the coarse mesh with large scale of collocation points is used in the method. Theoretical results show that when h approximate to H-3, eta(l, N) approximate to (eta(L, N))(3), the novel two-level discretization method achieves the same convergence accuracy in norm parallel to center dot parallel to(L rho 2(Gamma)circle times L2(D)) (L-rho(2)(Gamma) is the weighted L-2 space with rho a probability density function) as that for the original semilinear problem directly by sparse grid stochastic collocation method with T-h(D) and large scale collocation points eta(l, N) in random spaces.