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Compressive space-time Galerkin discretizations of parabolic partial differential equations

Preprint, 2015

We study linear parabolic initial-value problems in a space-time
variational formulation based on fractional calculus. This
formulation uses ``time derivatives of order one half'' on the
bi-infinite time axis. We show that for linear, parabolic
initial-boundary value problems on $(0,\infty)$, the corresponding
bilinear form admits an inf-sup condition with sparse tensor product
trial and test function spaces. We deduce optimality of
compressive, space-time Galerkin discretizations, where stability of
Galerkin approximations is implied by the well-posedness of the
parabolic operator equation. The variational setting adopted here
admits more general Riesz bases than previous work; in particular,
\emph{no stability in negative order Sobolev spaces on the spatial
or temporal domains} is required of the Riesz bases accommodated
by the present formulation. The trial and test spaces are based on
Sobolev spaces of equal order $1/2$ with respect to the temporal
variable. Sparse tensor products of multi-level decompositions of
the spatial and temporal spaces in Galerkin discretizations lead to
large, non-symmetric linear systems of equations. We prove that
their condition numbers are uniformly bounded with respect to the
discretization level. In terms of the total number of degrees of
freedom, the convergence orders equal, up to logarithmic terms,
those of best $N$-term approximations of solutions of the
corresponding elliptic problems.

Compressive Galerkin

Fractional Calculus

Adaptivity

Wavelets

Parabolic Problems

Space-Time Discretization