We propose a new adaptive mesh refinement (AMR) strategy, the goal is to reach a given error tolerance with the least amount of computational cost. This strategy is especially attractive in the setting of a first-order system least squares (FOSLS) finite element formulation in conjunction with algebraic multigrid (AMG) methods in the context of nested iteration (NI). To accomplish this, the refinement decisions are determined based on minimizing the predicted accuracy-per-computational-cost efficiency (ACE). The nested iteration approach produces a sequence of refinement levels in which the error is equally distributed across elements on a relatively coarse grid. Once the solution is numerically resolved, refinement becomes nearly uniform.

This talk will first describe the algorithm and demonstrate its efficiency on a simple test problem. Then, modifications that yield an efficient parallel algorithm will be discussed. These involve a geometric binning strategy to reduce communication cost. Load balancing begins at coarse levels using a parallel quad-tree and a space filling curve. We show that this automatically ameliorates load balancing issues at finer levels.

Numerical results are presented for various examples including a 2D Poisson problem with steep gradients and a 2D reduced model of the incompressible, resistive magnetohydrodynamic equations (MHD). We show that, by using the NI-FOSL-AMG-ACE approach, we are able to resolve the physics features using a small number of work units. Using Frost, The NCAR/CU Blue Gene/L supercomputer, we demonstrate excellent weak and strong scalability up to 4,096 processor for problems with up to 15 million elements.