@Article{Mitchell:2014:CAS,
  author =       "William F. Mitchell and Marjorie A. McClain",
  title =        "A Comparison of $hp$-Adaptive Strategies for Elliptic
                  Partial Differential Equations",
  journal =      "{ACM} Transactions on Mathematical Software",
  volume  =      41,
  number  =      1,
  accepted =     "3 January 2014",
  upcoming =     "true", 
  abstract =     "
                  The $hp$ version of the finite
                  element method ($hp$-FEM)
                  combined with adaptive mesh refinement
                  is a particularly efficient method
                  for solving PDEs because it can
                  achieve an exponential convergence
                  rate in the number of degrees of
                  freedom. $hp$-FEM allows for
                  refinement in both the element size,
                  $h$, and the polynomial degree, $p$. Like
                  adaptive refinement for the $h$ version
                  of the finite element method, \textit{a
                  posteriori} error estimates can be used
                  to determine where the mesh needs to be
                  refined, but a single error estimate can
                  not simultaneously determine whether it
                  is better to do the refinement by $h$ or
                  $p$. Several strategies for making this
                  determination have been proposed over the
                  years. These strategies are summarized,
                  and the results of a numerical
                  experiment to study the performance
                  of these strategies is presented. It
                  was found that the reference solution
                  based methods are very effective,
                  but also considerably more expensive,
                  in terms of computation time, than other
                  approaches. The method based on \textit{a
                  priori} knowledge is very effective when
                  there are known point singularities. The
                  method based on the decay rate of the
                  expansion coefficients appears to be
                  the best choice as a general strategy
                  across all categories of problems,
                  whereas many of the other strategies
                  perform well in particular situations
                  and are reasonable in general.",
}