Sparse Tensor Discretization of Elliptic sPDEs

We propose and analyze sparse deterministic-stochastic tensor Galerkin finite element methods (sparse sGFEMs) for the numerical solution of elliptic partial differential equations (PDEs) with random coefficients in a physical domain Dᵈ. In tensor product sGFEMs, the variational solution to the boundary value problem is approximated in tensor product finite element spaces V^Γ⊗ VD, where V^Γ and VD denote suitable finite dimensional subspaces of the stochastic and deterministic function spaces, respectively. These approaches lead to sGFEM algorithms of complexity O(N_Γ ND), where N_Γ= V^Γ and ND= VD. In this work, we use hierarchic sequences V^Γ₁⊂ V^Γ₂⊂… and VD₁⊂ VD₂⊂… of finite dimensional spaces to approximate the law of the random solution. The hierarchies of approximation spaces allow us to define sparse tensor product spaces V^Γ_⊗̂VD_, =1,2,, yielding algorithms of O(N_Γlog ND+NDlog N_Γ) work and memory. We estimate the convergence rate of sGFEM for algebraic decay of the input random field Karhunen–Loève coefficients. We give an algorithm for an input adapted a-priori selection of deterministic and stochastic discretization spaces. The convergence rate in terms of the total number of degrees of freedom of the proposed method is superior to Monte Carlo approximations. Numerical examples illustrate the theoretical results and demonstrate superiority of the sparse tensor product discretization proposed here versus the full tensor product approach.