We will develop the extensions to the layered AMR tools required to
support the applications specified above. This includes the fourth-order
Mehrstellen discretizations of Poisson's equation required to support
the method of local corrections
[4,3] and other
particle-particle particle-mesh
methods, and the coupling of those methods to embedded boundary
discretizations of irregular geometries. We will also begin to develop
latency-tolerant solvers for constant coefficient elliptic problems
along the lines of the approach used in
[5]. In this approach, one uses potential theory to
construct a domain decomposition method that requires only one iteration
between the local solves and the global coarse solve. This is
comparison to the more traditional iterative approaches, such as the
additive Schwartz procedure, that require many iterations between the
local and global solutions.