(joint with Dmitry Karpeev (Argonne National Laboratory))
The finite element paradigm relies on a decomposition of space into small pieces on which
function spaces can be explicitly constructed, followed by an assembly on the overlap of
the pieces to produce a global interpolant. This process is also familiar from the theory
of manifolds, as well as, sheaves. We have abstracted this relation, in analogy
with algebraic topology, to produce a flexible software framework for arbitrary element methods
which is parallel, scalable, dimension independent, and quite easy to manipulate.
This framework allows easy implementation of novel multilevel solver algorithms. As an example, we will discuss the Full Approximation Scheme, a nonlinear multigrid variant. It overcomes the well-known bandwidth bottleneck experiences by global Newton-based solvers. Finally, we will discuss how techniques from algebraic geometry might be used to improve both the performance and robustness of FAS applied to the Navier-Stokes equations.