Sparse finite element approximation of high-dimensional transport-dominated diffusion problems
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 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 $-a:\nabla \nabla u+b·\nabla u+cu=f\left(x\right)$, $x\in \Omega ={\left(0,1\right)}^{d}\subset {ℝ}^{d}$, where $a\in {ℝ}^{d×d}$ is a symmetric positive semidefinite matrix, using piecewise polynomials of degree $p\ge 1$. 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 ${u}_{h}$ on a partition of $\Omega$ of mesh size $h={h}_{L}={2}^{-L}$ satisfies the following bound in the streamline-diffusion norm $|||·{|||}_{\mathrm{SD}}$, provided $u$ belongs to the space ${ℋ}^{k+1}\left(\Omega \right)$ of functions with square-integrable mixed $\left(k+1\right)$st derivatives: $|||u-{u}_{h}{|||}_{\mathrm{SD}}\le {C}_{p,t}{d}^{2}max\left\{{\left(2-p\right)}_{+},{\kappa }_{0}^{d-1},{\kappa }_{1}^{d}\right\}\left(|\sqrt{a}|{h}_{L}^{t}+{|b|}^{\frac{1}{2}}{h}_{L}^{t+\frac{1}{2}}+{c}^{\frac{1}{2}}{h}_{L}^{t+1}{\phantom{\rule{-0.166667em}{0ex}}\right)|u|}_{{ℋ}^{t+1}\left(\Omega \right)},\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}$ where ${\kappa }_{i}={\kappa }_{i}\left(p,t,L\right)$, $i=0,1$, and $1\le t\le min\left(k,p\right)$. 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 ${\kappa }_{0},{\kappa }_{1}\in \left(0,1\right)$, and hence for $p\ge 2$ the ‘error constant’ ${C}_{p,t}{d}^{2}max\left\{{\left(2-p\right)}_{+},{\kappa }_{0}^{d-1},{\kappa }_{1}^{d}\right\}$ exhibits exponential decay as $d\to \infty$; in the case of a general symmetric positive semidefinite matrix $a$, the error constant is shown to grow no faster than $𝒪\left({d}^{2}\right)$. In any case, in the absence of assumptions that relate $L$, $p$ and $d$, the error $|||u-{u}_{h}{|||}_{\mathrm{SD}}$ is still bounded by ${\kappa }_{*}^{d-1}|{log}_{2}{h}_{L}{|}^{d-1}𝒪\left(|\sqrt{a}|{h}_{L}^{t}+{|b|}^{\frac{1}{2}}{h}_{L}^{t+\frac{1}{2}}+{c}^{\frac{1}{2}}{h}_{L}^{t+1}\right)$, where ${\kappa }_{*}\in \left(0,1\right)$ for all $L,p,d\ge 2$.

DOI : https://doi.org/10.1051/m2an:2008027
Classification:  65N30
Keywords: high-dimensional Fokker-Planck equations, partial differential equations with nonnegative characteristic form, sparse finite element method
@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://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, Volume 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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