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{\large Jacob Schroder \\
{\bf Smoothed Aggregation Multigrid for One-Dimensional Helmholtz Problems }}

Siebel Center for Computer Science \\ 4328 Siebel Center \\ 201 N Goodwin Ave \\ Urbana \\ IL 61801 \\ USA
\\
{\tt jacob.bb.schroder@gmail.com}\\
Luke Olson\\
{\em Assistant Professor, Computer Science Department, University of Illinois at Urbana-Champaign}\end{center}

We present a purely algebraic smoothed aggregation (SA) multigrid method
for one-dimensional scalar Helmholtz problems with exterior radiation
boundary conditions. The scalar Helmholtz problem is particularly
difficult for algebraic multigrid (AMG) solvers. Not only can the
discrete operator be indefinite and non-self-adjoint, but it also allows
for oscillatory error components that yield relatively small residuals.
These oscillatory error components are not effectively handled by either
standard relaxation or standard coarsening procedures. We address these
difficulties through modifications of the SA method and by providing the
setup phase with appropriate wave-like near-nullspace candidates. Our
results for GMRES preconditioned with the proposed SA method exhibit near
grid-independent performance, along with consistent performance across a
wide range of frequencies.


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