  Important computational physics problems are often large-scale in nature, and
it is highly desirable to have robust and high performing computational
frameworks that can quickly address these problems. However, it is no trivial
task to determine whether a computational framework is performing efficiently
or is scalable. The aim of this paper is to present various strategies for
better understanding the performance of any parallel computational frameworks
for solving PDEs. Important performance issues that negatively impact
time-to-solution are discussed, and we propose a performance spectrum analysis
that can enhance one's understanding of critical aforementioned performance
issues. As proof of concept, we examine commonly used finite element simulation
packages and software and apply the performance spectrum to quickly analyze the
performance and scalability across various hardware platforms, software
implementations, and numerical discretizations. It is shown that the proposed
performance spectrum is a versatile performance model that is not only
extendable to more complex PDEs such as hydrostatic ice sheet flow equations,
but also useful for understanding hardware performance in a massively parallel
computing environment. Potential applications and future extensions of this
work are also discussed.
