
Dual Functions for Parallel Adaptive Methods
Jeffrey Ovall (MPI Leipzig)
In 2000 Bank and Holst proposed a new paradigm for parallel adaptive
meshing algorithms
[SIAM J. Sci. Comp., 22 (4), 2000;
SIAM Review, 45 (2), 2003]. The load balancing problem, which is
problematic for many algorithms, is addressed by solving a small
elliptic problem on a single processor (or independantly on all
processors), and using a posteriori error estimates to partition the
domain. Each processor then independently performs several
iterations of solveestimaterefine on the entire problem until
a target number of vertices is reached, with the error estimates
outside of its assigned subdomain artificially damped by 10^{6}.
At the end of this second stage a global conforming mesh is formed
from the fine contributions from each processor, and a final global
parallel iterative solve done on this mesh with the initial guess
build from the fine solutions on each processor.
The ability of the second stage to provide an adequate setup for the
final global solve is strained somewhat by the presence of
singularities or other phenomena which have an unusually strong
influence on global behavior. We propose a different error estimate
weighting scheme which is based on the solution of an appropriate dual
problem, and which better takes into account global influences on
local behavior.

