Speaker
Description
An algebraic multigrid (AMG) algorithm is proposed for curl-curl electro-magnetics PDEs that are discretized with 1st order edge elements. The key idea behind the algorithm centers on generating edge interpolation operators such that certain near null space properties of the discrete curl-curl operator are preserved on coarse levels so that good AMG convergence rates can be obtained. Specifically, the algorithm guarantees that the algebraically constructed discrete gradient operator lies in the null space of the algebraically generated discrete curl-curl operator on all grid levels of the multigrid hierarchy.
The new algorithm is based on a varient of energy minimization AMG. It takes an already-computed nodal interpolation operator to then define an edge interpolant that satisfies a related discrete commuting relationship while minimizing the energy of the edge interpolation basis functions. Unlike previous works, this algorithm is general in that it can essentially transform any standard nodal interpolation operator into an edge interpolation operator that is suitable for curl-curl problems. Thus, it is possible to leverage traditional AMG schemes to first generate nodal interpolation operators which can then be adapted to edge interpolation operators. We show that the new algorithm can produce "ideal geometric" edge interpolants in some limited cases. Numerical results are provided showing the overall efficacy, comparing with traditional Reitzinger and Schoberl schemes.