  Numerical solution of partial differential equations on parallel computers
using domain decomposition usually requires synchronization and communication
among the processors. These operations often have a significant overhead in
terms of time and energy. In this paper, we propose communication-efficient
parallel algorithms for solving partial differential equations that alleviate
this overhead. First, we describe an asynchronous algorithm that removes the
requirement of synchronization and checks for termination in a distributed
fashion while maintaining the provision to restart iterations if necessary.
Then, we build on the asynchronous algorithm to propose an event-triggered
communication algorithm that communicates the boundary values to neighboring
processors only at certain iterations, thereby reducing the number of messages
while maintaining similar accuracy of solution. We demonstrate our algorithms
on a successive over-relaxation solver for the Pressure Poisson equation
arising from variable density incompressible multiphase flows in 3-D and show
that our algorithms improve time and energy efficiency.
