Advection-diffusion PDEs are prevalent in models of many physical applications in science and engineering. In this talk, we focus on a scalar-valued advection-diffusion-reaction equation of the form
where
Domain decomposition methods display nearly optimal parallel scalability within the advection-dominated regime, but typically do not scale well for diffusion-dominated problems. The converse occurs when using multigrid methods. Our goal is to find one method that is scalable in both regimes, through using a hybrid approach combining restrictive additive Schwarz (domain decomposition) and geometric multigrid.
Our approach begins with a Schur complement formulation of the
linearized implicit system (
) as with the FETI (Farhat and
Roux, 1991), BDD (Mandel, 1993), and BDDC (Dohrmann, 2003) algorithms, where the
unknowns
are split into two sets: those residing in a domain
interior,
, and those residing at inter-processor boundaries,
. With this decomposition, one may similarly decompose the
full Jacobian matrix into four associated blocks,
Since the
where
While traditional domain-decomposition methods do not solve this
global Schur complement system directly (and instead solve an
interface system based on only a small fraction of unknowns), we solve
the full interface system using a multilevel technique similar to
multigrid. Here, our fine grid problem consists of the entire
Schur complement (interface) system. We then proceed through a
traditional set of V-cycle iterations, where at each level we obtain
an increasingly coarse subset of the full interface system. However,
due to the
term in
, all residual corrections are
evaluated in a matrix-free fashion, using FGMRES as a smoother.
After presenting the details of our algorithm, we present simulation results using the Ranger supercomputer at TACC. We investigate problems in both the advection- and diffusion-dominated regimes, and examine scalability of both iteration count and wall-clock time.