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Issue ESAIM: M2AN
Volume 42, Number 5, September-October 2008
Page(s) 777 - 819
DOI 10.1051/m2an:2008027
Published online 30 July 2008

ESAIM: M2AN 42 (2008) 777-819
DOI: 10.1051/m2an:2008027

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

Christoph Schwab1, Endre Süli2 and Radu Alexandru Todor1

1  Seminar für Angewandte Mathematik, Eidgenössische Technische Hochschule, 8092 Zürich, Switzerland. schwab@sam.math.ethz.ch; todor@math.ethz.ch
2  University of Oxford, Computing Laboratory, Wolfson Building, Parks Road, Oxford OX1 3QD, UK. endre.suli@comlab.ox.ac.uk

Received March 7, 2007. Received February 11, 2008. Published online July 30, 2008.

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 $p\geq 1$. 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 $\Omega$ of mesh size h=hL = 2-L satisfies the following bound in the streamline-diffusion norm $\vert\vert\vert\cdot\vert\vert\vert _{\rm SD}$, provided u belongs to the space $\mathcal{H}^{k+1}(\Omega)$ of functions with square-integrable mixed (k+1)st derivatives:

\begin{displaymath}\vert\vert\vert u-u_h\vert\vert\vert _{\rm SD}\leq C_{p,t} d^... ...vert u\vert _{\mathcal{H}^{t+1}(\Omega)}, \qquad \qquad \qquad \end{displaymath}

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 $p \geq 2$ the `error constant' $C_{p,t} d^2 \max\{(2-p)_+,\kappa_0^{d-1},\kappa_1^d\}$ exhibits exponential decay as $d \rightarrow \infty$; 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 $\vert\vert\vert u - u_h\vert\vert\vert _{\rm SD}$ is still bounded by $\kappa_\ast^{d-1} \vert\log_2 h_L\vert^{d-1}\mathcal{O}(\vert\sqrt{a}\vert h_L^t + \vert b\vert^{\frac} h_L^{t+\frac} + c^{\frac} h_L^{t+1})$, where $\kappa_\ast \in (0,1)$ for all $L, p, d \geq 2$.


Mathematics Subject Classification. 65N30.

Key words: High-dimensional Fokker-Planck equations, partial differential equations with nonnegative characteristic form, sparse finite element method.


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