  Plasma-based acceleration (PBA) has emerged as a promising candidate for the
accelerator technology used to build a future linear collider and/or an
advanced light source. In PBA, a trailing or witness particle beam is
accelerated in the plasma wave wakefield (WF) created by a laser or particle
beam driver. The distance over which the drive beam evolves is several orders
of magnitude larger than the wake wavelength. This large disparity in length
scales is amenable to the quasi-static approach. Three-dimensional (3D),
quasi-static (QS), particle-in-cell (PIC) codes, e.g., QuickPIC, have been
shown to provide high fidelity simulation capability with 2-4 orders of
magnitude speedup over 3D fully explicit PIC codes. We describe a mesh
refinement scheme that has been implemented into the 3D QS PIC code, QuickPIC.
We use a very fine (high) resolution in a small spatial region that includes
the witness beam and progressively coarser resolutions in the rest of the
simulation domain. A fast multigrid Poisson solver has been implemented for the
field solve on the refined meshes and a Fast Fourier Transform (FFT) based
Poisson solver is used for the coarse mesh. The code has been parallelized with
both MPI and OpenMP, and the parallel scalability has also been improved by
using pipelining. A preliminary adaptive mesh refinement technique is described
to optimize the computational time for simulations with an evolving witness
beam size. Several test problems are used to verify that the mesh refinement
algorithm provides accurate results. The results are also compared to highly
resolved simulations with near azimuthal symmetry using a new hybrid QS PIC
code QPAD that uses a PIC description in the coordinates ($r$, $ct-z$) and a
gridless description in the azimuthal angle, $\phi$.
