Sparse Finite Element Approximation of High-dimensional Transport-dominated Diffusion Problems

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 −a : ∇∇u + b · ∇u + cu = f(x), x ∈ Ω = (0, 1) ⊂ R, where a ∈ Rd×d 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 uh on 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 H(Ω) of functions with square-integrable mixed (k + 1)st derivatives: |||u− uh|||SD ≤ Cp,tdmax{(2− p)+, κ 0 , κ d 1}(| √ a|hL + |b| 1 2 h t+ 2 L + c 1 2 h L )|u|Ht+1(Ω), where κi = κi(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 κ0, κ1 ∈ (0, 1), and hence for p ≥ 2 the ‘error constant’ Cp,tdmax{(2 − p)+, κd−1 0 , κ1} exhibits exponential decay as d→∞; in the case of a general symmetric positive semidefinite matrix a, the error constant is shown to grow no faster than O(d). In any case, in the absence of assumptions that relate L, p and d, the error |||u− uh|||SD is still bounded by κd−1 ∗ | log2 hL|O(| √ a|hL + |b| 1 2 h t+ 2 L + c 1 2 h L ), where κ∗ ∈ (0, 1) for all L, p, d ≥ 2. 1991 Mathematics Subject Classification. 65N30. The dates will be set by the publisher.