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Computer Science Department

Computer Science Department

Algebraic Multigrid and a Compatible Gauge Reformulation of Maxwell's Equations

Chris Siefert
Sandia National Laboratory
Fri, Nov 30, 3 PM, McGl 020

With the rise in popularity of compatible finite element, finite difference and finite volume discretizations for the time domain eddy current equations, there has been a corresponding need for fast solvers of the resulting linear algebraic systems. However, the traits that make compatible discretizations a preferred choice for the Maxwell's equations also render these linear systems essentially intractable by truly black-box techniques. We propose a new algebraic reformulation of the discrete eddy current equations along with a new algebraic multigrid technique (AMG) for this reformulated problem. The reformulation process takes advantage of a discrete Hodge decomposition to replace the discrete eddy current equations by an equivalent 2x2 block linear system.

While this new AMG technique requires somewhat specialized treatment on the finest mesh, the coarser meshes can be handled using standard methods for Laplace-type problems, allowing for code reuse. Our new AMG method is applicable to a wide range of compatible methods on structured and unstructured grids. We illustrate the new technique, using edge elements in the context of smoothed aggregation AMG. We present computational results on test problems in two and three dimensions, as well as scaling results from our Z-Pinch simulation code at up to 5,832 processors.