Next I will discuss ongoing work investigating AMG for this application. Because the coarse pressure solver is invoked tens of thousands of times for the same equation, we can afford an almost arbitrarily long time spent in the set-up phase of AMG. I am investigating how this fact can be leveraged to tune the invocation stage to be as fast and efficient as possible.
Past investigators have considered using Sparse Approximate Inverses (SAI) as smoothers in multigrid algorithms. They are attractive because (1) their application is a matrix-vector product and hence inherently parallel, (2) they are inexpensive to compute, and (3) for a given sparsity pattern they are parameter-free. Building on the theory of Falgout, we will show how the SAI idea can be used also to cheaply construct an optimal sparse interpolation operator. We will show that the "residual" of this operator, against the optimal dense interpolation operator, can be used to easily identify those fine degrees of freedom which cannot be adequately interpolated (and hence should be added to the coarse set). This procedure is both simpler and less ad-hoc than coarsening via compatible relaxation. I will present a new compatible relaxation scheme which yields the exact spectrum of the full two-level method while remaining computable, and show that it might be used to improve the selection of an SAI smoother.