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 u
h
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.