  In this paper, we compute $\epsilon$-approximate Nash equilibria in atomic
splittable polymatroid congestion games with convex Lipschitz continuous cost
functions. The main approach relies on computing a pure Nash equilibrium for an
associated integrally-splittable congestion game, where players can only split
their demand in integral multiples of a common packet size. It is known that
one can compute pure Nash equilibria for integrally-splittable congestion games
within a running time that is pseudo-polynomial in the aggregated demand of the
players. As the main contribution of this paper, we decide for every
$\epsilon>0$, a packet size $k_{\epsilon}$ and prove that the associated
$k_{\epsilon}$-splittable Nash equilibrium is an $\epsilon$-approximate Nash
equilibrium for the original game.
  We further show that our result applies to multimarket oligopolies with
decreasing, concave Lipschitz continuous price functions and quadratic
production costs: there is a polynomial time transformation to atomic
splittable polymatroid congestion games implying that we can compute
$\epsilon$-approximate Cournot-Nash equilibria within pseudo-polynomial time.
