  In this paper, we develop a new parallel auxiliary grid algebraic multigrid
(AMG) method to leverage the power of graphic processing units (GPUs). In the
construction of the hierarchical coarse grid, we use a simple and fixed
coarsening procedure based on a region quadtree generated from an auxiliary
grid. This allows us to explicitly control the sparsity patterns and operator
complexities of the AMG solver. This feature provides (nearly) optimal load
balancing and predictable communication patterns, which makes our new algorithm
suitable for parallel computing, especially on GPU. We also design a parallel
smoother based on the special coloring of the quadtree to accelerate the
convergence rate and improve the parallel performance of this solver. Based on
the CUDA toolkit [40], we implemented our new parallel auxiliary grid AMG
method on GPU and the numerical results of this implementation demonstrate the
efficiency of our new method. The results achieve an average speedup of over 4
on quasi-uniform grids and 2 on shape regular grids when compared to the AMG
implementation in CUSP.
