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

Schwab, Christoph; Süli, Endre; Todor, Radu Alexandru

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 $-a:\nabla \nabla u+b·\nabla u+cu=f\left(x\right)$, $x\in \Omega ={(0,1)}^d\subset {ℝ}^d$, where $a\in {ℝ}^{d\times d}$ 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 $\left|\right||·{\left|\right||}_{\mathrm{SD}}$, provided $u$ belongs to the space ${\mathscr{H}}^{k+1}\left(\Omega \right)$ of functions with square-integrable mixed $(k+1)$st derivatives:

$$\left|\right||u-u_h{\left|\right||}_{\mathrm{SD}}\le C_{p,t}{d}^{2}max\{{(2-p)}_{+},{\kappa }_0^{d-1},{\kappa }_1^d\}\left(\right|\sqrt{a}|h_L^t+{\left|b\right|}^{\frac{1}{2}}{h}_L^{t+\frac{1}{2}}+c^{\frac{1}{2}}{h}_L^{t+1}{\left)\right|u|}_{{\mathscr{H}}^{t+1}\left(\Omega \right)},$$

where ${\kappa }_i={\kappa }_i(p,t,L)$, $i=0,1$, and $1\le t\le min(k,p)$. 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 $p\ge 2$ the ‘error constant’ $C_{p,t}{d}^{2}max\{{(2-p)}_{+},{\kappa }_0^{d-1},{\kappa }_1^d\}$ 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 $\left|\right||u-u_h{\left|\right||}_{\mathrm{SD}}$ is still bounded by ${\kappa }_{*}^{d-1}|{log}_2{h}_L{|}^{d-1}𝒪\left(\right|\sqrt{a}|h_L^t+{\left|b\right|}^{\frac{1}{2}}{h}_L^{t+\frac{1}{2}}+c^{\frac{1}{2}}{h}_L^{t+1})$, where ${\kappa }_{*}\in (0,1)$ for all $L,p,d\ge 2$.

Classification : 65N30
Mots clés : high-dimensional Fokker-Planck equations, partial differential equations with nonnegative characteristic form, sparse finite element method

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