MDFEM: Multivariate decomposition finite element methods for elliptic PDEs with lognormal diffusion coefficients using higher-order QMC and FEM
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We introduce the novel multivariate decomposition finite element method (MDFEM) for solving elliptic PDEs with lognormal diffusion coefficients, that is, when the diffusion coefficient a has the form a=(Z) where Z is a Gaussian random field defined as Z(y) = ∑_j ≥ 1 y_j ϕ_j with y_j ∼N(0,1) and a sequence of functions {ϕ_j}_j ≥ 1. We estimate the expected value of some linear functional of the solution as an infinite-dimensional integral over the parameter space. The proposed algorithm combines the multivariate decomposition method (MDM), to compute infinite-dimensional integrals, with the finite element method (FEM), to solve different instances of the PDE. This allows us to apply higher-order multivariate quadrature methods for integration over the Euclidean space with respect to the Gaussian distribution, and, hence, considerably improves upon existing results which only use first order cubature rules. We develop multivariate quadrature methods for integration over the finite-dimensional Euclidean space with respect to the Gaussian distribution. By linear transformations of interlaced polynomial lattice rules from a unit cube to a multivariate box of the Euclidean space we achieve higher-order convergence rates for functions belonging to a class of anchored Gaussian Sobolev spaces. These cubature rules are then used in the MDFEM algorithm. Under appropriate conditions, applying higher-order cubature rules we achieve higher-order convergence rates in term of error vs cost, i.e., the computational cost to achieve an accuracy of O(ϵ) is O(ϵ^-(1+d/τ)p^*/(1-p^*) -d/τ) where d is the physical dimension, τ is the convergence rate of the finite element method and p^* represents the sparsity of {ϕ_j}_j ≥ 1.
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