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{\large James Lai \\
{\bf Algebraic multigrid for high order hierarchical edge elements.}}

University of Illinois at Urbana-Champaign \\ Department of Computer Science \\ 201 N Goodwin Ave \\ Urbana \\ IL 61801
\\
{\tt jhlai2@illinois.edu}\\
Luke Olson\end{center}

We propose an algebraic multigrid (AMG) method for systems discretized by
high order $H$(curl) elements. Standard AMG methods perform poorly on
these problems because of the large null space of the curl operator. We
use hierarchical high order $H$(curl) conforming elements. Our algorithm
combines ideas from $p$-multigrid and AMG for edge elements to obtain a
high order AMG method. The initial levels in our hierarchy are
constructed by discretizing the problem for decreasing orders of $p$.
Since we use a hierarchical basis, the integrid transfer operators are
trivial. To address complexity issues, we do not visit every $p$ in our
hierarchy. After $p=0$ is reached, we apply smoothed aggregation AMG for
edge elements. We conclude with numerical results for the proposed AMG
algorithm applied to the 2D eddy current problem for bases up to order
$p=9$.


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