  Researchers have employed variations of the Smoluchowski coagulation equation
to model a wide variety of both organic and inorganic phenomena and with
relatively few known analytical solutions, numerical solutions play an
important role in studying this equation. In this article, we consider
numerical approximations, focusing on how different discretization schemes
impact the accuracy of approximate solution moments. Pursuing the eventual goal
of comparing simulated solutions to experimental data, we must carefully choose
the numerical method most appropriate to the type of data we attain. Within
this context, we compare and contrast the accuracy and computational cost of a
finite element approach and a finite volume-based scheme.
  Our study provides theoretical and numerical evidence that the finite element
approach achieves much more accuracy when the system aggregates slowly, and it
does so with much less computation cost. Conversely, the finite volume method
is slightly more accurate approximating the zeroth moment when the system
aggregates quickly and is much more accurate approximating the first moment in
general.
  Lastly, our study also provides numerical evidence that the finite element
method (conventionally considered first order) actually belongs to a class of
discontinuous Galerkin methods that exhibit superconvergence, or second order
in our case.
