  Thermal convection driven by internal heat sources and sinks was recently
shown experimentally to exhibit the mixing-length, or "ultimate",
scaling-regime: the Nusselt number $Nu$ (dimensionless heat flux) increases as
the square-root of the Rayleigh-number $Ra$ (dimensionless internal temperature
difference). While for standard Rayleigh-B\'enard convection this scaling
regime was proven to be a rigorous upper bound on the Nusselt number, we show
that this is not so for convection driven by internal sources and sinks. To
wit, we introduce an asymptotic expansion to derive steady nonlinear solutions
in the limit of large $Ra_Q$, the Rayleigh-number based on the strength of the
heat source. We illustrate this procedure for a simple sinusoidal heat source
and show that it achieves heat transport enhancement beyond the mixing-length
scaling regime: $Nu$ increases linearly with $Ra$ over this branch of
solutions. Using rigorous upper bound theory, we prove that the scaling regime
$Nu \sim Ra$ of the asymptotic solution corresponds to a maximization of the
heat flux subject to simple dynamical constraints, up to a dimensionless
prefactor. Not only do 2D numerical simulations confirm the analytical solution
for the sinusoidal source, but, more surprisingly, they indicate that it is
stable and indeed achieved by the system up to the highest $Ra_Q$ investigated
numerically, with a heat transport efficiency orders of magnitude higher than
the standard mixing-length estimate.
