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

Christoph Schwab; Endre Süli; Radu Alexandru Todor

ESAIM: Mathematical Modelling and Numerical Analysis (2008)

  • Volume: 42, Issue: 5, page 777-819
  • ISSN: 0764-583X

Abstract

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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 + c u = f ( x ) , x Ω = ( 0 , 1 ) d d , where a d × 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 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 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 κ 0 , κ 1 ( 0 , 1 ) , and hence for p ≥ 1 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, d ≥ 2.

How to cite

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Schwab, Christoph, Süli, Endre, and Todor, Radu Alexandru. "Sparse finite element approximation of high-dimensional transport-dominated diffusion problems." ESAIM: Mathematical Modelling and Numerical Analysis 42.5 (2008): 777-819. <http://eudml.org/doc/250411>.

@article{Schwab2008,
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 $-a:\nabla\nabla u + b \cdot \nabla u + cu = f(x)$, $x \in \Omega = (0,1)^d \subset \mathbb\{R\}^d$, where $a \in \mathbb\{R\}^\{d\times d\}$ 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 $|||\cdot|||_\{\rm SD\}$, provided u belongs to the space $\mathcal\{H\}^\{k+1\}(\Omega)$ of functions with square-integrable mixed (k+1)st derivatives: \[ |||u-u\_h|||\_\{\rm SD\}\leq C\_\{p,t\} d^2 \max\\{(2-p)\_+,\kappa\_0^\{d-1\},\kappa\_1^d\\} (|\sqrt\{a\}| h\_L^t + |b|^\{\frac\{1\}\{2\}\} h\_L^\{t+\frac\{1\}\{2\}\} + c^\{\frac\{1\}\{2\}\} h\_L^\{t+1\} \!)|u|\_\{\mathcal\{H\}^\{t+1\}(\Omega)\}, \qquad \qquad \qquad \] where $\kappa_i=\kappa_i(p,t,L)$, i=0,1, and $1 \leq t \leq \min(k,p)$. 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 $\kappa_0, \kappa_1 \in (0,1)$, and hence for p ≥ 1 the 'error constant' $C_\{p,t\} d^2 \max\\{(2-p)_+,\kappa_0^\{d-1\},\kappa_1^d\\}$ 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 $\mathcal\{O\}(d^2)$. In any case, in the absence of assumptions that relate L, p and d, the error $|||u - u_h|||_\{\rm SD\}$ is still bounded by $\kappa_\ast^\{d-1\} |\log_2 h_L|^\{d-1\}\mathcal\{O\}(|\sqrt\{a\}| h_L^t + |b|^\{\frac\{1\}\{2\}\} h_L^\{t+\frac\{1\}\{2\}\} + c^\{\frac\{1\}\{2\}\} h_L^\{t+1\})$, where $\kappa_\ast \in (0,1)$ for all L, p, d ≥ 2. },
author = {Schwab, Christoph, Süli, Endre, Todor, Radu Alexandru},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {High-dimensional Fokker-Planck equations; partial differential equations with nonnegative characteristic form; sparse finite element method.; high-dimensional Fokker-Planck equations; non-negative characteristic form; sparse finite element method; error bounds; convergence; sparse tensor-product spaces},
language = {eng},
month = {7},
number = {5},
pages = {777-819},
publisher = {EDP Sciences},
title = {Sparse finite element approximation of high-dimensional transport-dominated diffusion problems},
url = {http://eudml.org/doc/250411},
volume = {42},
year = {2008},
}

TY - JOUR
AU - Schwab, Christoph
AU - Süli, Endre
AU - Todor, Radu Alexandru
TI - Sparse finite element approximation of high-dimensional transport-dominated diffusion problems
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2008/7//
PB - EDP Sciences
VL - 42
IS - 5
SP - 777
EP - 819
AB - 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 $-a:\nabla\nabla u + b \cdot \nabla u + cu = f(x)$, $x \in \Omega = (0,1)^d \subset \mathbb{R}^d$, where $a \in \mathbb{R}^{d\times d}$ 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 $|||\cdot|||_{\rm SD}$, provided u belongs to the space $\mathcal{H}^{k+1}(\Omega)$ of functions with square-integrable mixed (k+1)st derivatives: \[ |||u-u_h|||_{\rm SD}\leq C_{p,t} d^2 \max\{(2-p)_+,\kappa_0^{d-1},\kappa_1^d\} (|\sqrt{a}| h_L^t + |b|^{\frac{1}{2}} h_L^{t+\frac{1}{2}} + c^{\frac{1}{2}} h_L^{t+1} \!)|u|_{\mathcal{H}^{t+1}(\Omega)}, \qquad \qquad \qquad \] where $\kappa_i=\kappa_i(p,t,L)$, i=0,1, and $1 \leq t \leq \min(k,p)$. 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 $\kappa_0, \kappa_1 \in (0,1)$, and hence for p ≥ 1 the 'error constant' $C_{p,t} d^2 \max\{(2-p)_+,\kappa_0^{d-1},\kappa_1^d\}$ 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 $\mathcal{O}(d^2)$. In any case, in the absence of assumptions that relate L, p and d, the error $|||u - u_h|||_{\rm SD}$ is still bounded by $\kappa_\ast^{d-1} |\log_2 h_L|^{d-1}\mathcal{O}(|\sqrt{a}| h_L^t + |b|^{\frac{1}{2}} h_L^{t+\frac{1}{2}} + c^{\frac{1}{2}} h_L^{t+1})$, where $\kappa_\ast \in (0,1)$ for all L, p, d ≥ 2.
LA - eng
KW - High-dimensional Fokker-Planck equations; partial differential equations with nonnegative characteristic form; sparse finite element method.; high-dimensional Fokker-Planck equations; non-negative characteristic form; sparse finite element method; error bounds; convergence; sparse tensor-product spaces
UR - http://eudml.org/doc/250411
ER -

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