AbstractIn 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 {\mathcal{T}_{H}(D)} with small scale of sparse collocation points {\eta(L,N)} and solve a linearized equations on a fine mesh {\mathcal{T}_{h}(D)} using large scale of sparse collocation points {\eta(\ell,N)} (where {\eta(L,N),\eta(\ell,N)} are the numbers of sparse grid with respect to different levels {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\approx H^{3},\eta(\ell,N)\approx(\eta(L,N))^{3}}}, the novel two-level discretization method achieves the same convergence accuracy in norm {\|\cdot\|_{\mathcal{L}_{\rho}^{2}(\Gamma)\otimes\mathcal{L}^{2}(D)}} ({\mathcal{L}_{\rho}^{2}(\Gamma)} is the weighted {\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 {\mathcal{T}_{h}(D)} and large scale collocation points {\eta(\ell,N)} in random spaces.
An adaptive sparse-grid high-order stochastic collocation method for Bayesian inference in groundwater reactive transport modeling
