Sparse finite element approximation of high-dimensional transport-dominated diffusion problems
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 42 (2008) no. 5, p. 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.
@article{M2AN_2008__42_5_777_0,
     author = {Schwab, Christoph and S\"uli, Endre and Todor, Radu Alexandru},
     title = {Sparse finite element approximation of high-dimensional transport-dominated diffusion problems},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {42},
     number = {5},
     year = {2008},
     pages = {777-819},
     doi = {10.1051/m2an:2008027},
     zbl = {1159.65094},
     mrnumber = {2454623},
     language = {en},
     url = {http://http://www.numdam.org/item/M2AN_2008__42_5_777_0}
}
Schwab, Christoph; Süli, Endre; Todor, Radu Alexandru. Sparse finite element approximation of high-dimensional transport-dominated diffusion problems. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 42 (2008) no. 5, pp. 777-819. doi : 10.1051/m2an:2008027. http://www.numdam.org/item/M2AN_2008__42_5_777_0/

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