Sparse finite element approximation of high-dimensional transport-dominated diffusion problems
ESAIM: Modélisation mathématique et analyse numérique, Tome 42 (2008) no. 5, pp. 777-819.

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), xΩ=(0,1) d d , where a d×d is a symmetric positive semidefinite matrix, using piecewise polynomials of degree p1. 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 h on a partition of Ω of mesh size h=h L =2 -L satisfies the following bound in the streamline-diffusion norm |||·||| SD , provided u belongs to the space k+1 (Ω) of functions with square-integrable mixed (k+1)st derivatives:

|||u-u h ||| SD C p,t d 2 max{(2-p) + ,κ 0 d-1 ,κ 1 d }(|a|h L t +|b| 1 2 h L t+1 2 +c 1 2 h L t+1 )|u| t+1 (Ω) ,
where κ i =κ i (p,t,L), i=0,1, and 1tmin(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 κ 0 ,κ 1 (0,1), and hence for p2 the ‘error constant’ C p,t d 2 max{(2-p) + ,κ 0 d-1 ,κ 1 d } 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 2 ). In any case, in the absence of assumptions that relate L, p and d, the error |||u-u h ||| SD is still bounded by κ * d-1 |log 2 h L | d-1 𝒪(|a|h L t +|b| 1 2 h L t+1 2 +c 1 2 h L t+1 ), where κ * (0,1) for all L,p,d2.

DOI : 10.1051/m2an:2008027
Classification : 65N30
Mots clés : high-dimensional Fokker-Planck equations, partial differential equations with nonnegative characteristic form, sparse finite element method
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     title = {Sparse finite element approximation of high-dimensional transport-dominated diffusion problems},
     journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
     pages = {777--819},
     publisher = {EDP-Sciences},
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Schwab, Christoph; Süli, Endre; Todor, Radu Alexandru. Sparse finite element approximation of high-dimensional transport-dominated diffusion problems. ESAIM: Modélisation mathématique et analyse numérique, Tome 42 (2008) no. 5, pp. 777-819. doi : 10.1051/m2an:2008027. http://www.numdam.org/articles/10.1051/m2an:2008027/

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