### Abstract

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 𝒯H(D){\mathcal{T}_{H}(D)} with small scale of sparse collocation points η(L,N){\eta(L,N)} and solve a linearized equations on a fine mesh 𝒯h(D){\mathcal{T}_{h}(D)} using large scale of sparse collocation points η(ℓ,N){\eta(\ell,N)} (where η(L,N),η(ℓ,N){\eta(L,N),\eta(\ell,N)} are the numbers of sparse grid with respect to different levels L,ℓ{L,\ell} 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≈H3,η(ℓ,N)≈(η(L,N))3{{h\approx H^{3},\eta(\ell,N)\approx(\eta(L,N))^{3}}}, the novel two-level discretization method achieves the same convergence accuracy in norm ∥⋅∥ℒρ2(Γ)⊗ℒ2(D){\|\cdot\|_{\mathcal{L}_{\rho}^{2}(\Gamma)\otimes\mathcal{L}^{2}(D)}} (ℒρ2(Γ){\mathcal{L}_{\rho}^{2}(\Gamma)} is the weighted ℒ2{\mathcal{L}^{2}} space with ρ a probability density function) as that for the original semilinear problem directly by sparse grid stochastic collocation method with 𝒯h(D){\mathcal{T}_{h}(D)} and large scale collocation points η(ℓ,N){\eta(\ell,N)} in random spaces.