Sparse finite element approximation of high-dimensional transport-dominated diffusion problems

Abstract

We develop the analysis of stabilized sparse tensor-product finite element methods for high-dimensional, non-self-adjoint and possibly degenerate second-order partial differential equations of the form ΔΔu + b · Δu + cu = f(x), x ε ω = (0, 1)d ⊃ Rd, where a ε Rd×d where is a symmetric positive semidefinite matrix, using piecewise polynomials of degree p ≥ 1. Our convergence analysis is based on new high-dimensional approximation results in sparse tensor-product spaces. We show that the error between the analytical solution u and its stabilized sparse finite element approximation u hon a partition of ω of mesh size h = hL = 2 -L satisfies the following bound in the streamline-diffusion norm ||| · |||SD, provided u belongs to the space Hk+1(ω) of functions with square-integrable mixed (k+1)st derivatives: |||u - u h|||SD ≤ Cp,td2 max {(2 - p)+, k0d-1, k1d-1} (|√a|h Lt + |b|1/2hLt+1/2 + c1/2hLt+1)|u|ℋt+1(ω) where ki = ki(p,t,L), i = 0, 1, and 1 ≤ t ≤ min(k,p). We show, under various mild conditions relating L to p, L to d, or p to d, that in the case of elliptic transport-dominated diffusion problems k0, k1 ε (0,1), and hence for p ≥ 2 the 'error constant' Cp,td2 max {(2 - p)+, k0d-1, k1d} exhibits exponential decay as d → ∞ in the case of a general symmetric positive semidefinite matrix α, the error constant is shown to grow no faster than O(d2). In any case, in the absence of assumptions that relate L, p and d, the error |||u - uh|||SD is still bounded by k*d-1 | log2 hL |d-1 O(|√a|hLt + |b|1/2 h Lt+1/2 + c1/2 hLt+1/2), where k* ε (0,1) for all L,p,d ≤ 2. © 2008 EDP Sciences SMAI.