Algebraic Multigrid for H(curl) and H(div)

Joachim Schoeberl

Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368

Stefan Reizinger

SFB013 "Scientific Computing", University Linz, 4020 Linz, Austria


There exists a variety of algebraic multigrid methods for elliptic problems in Sobolev spaces. Recently, Arnold,Falk and Winther and Hiptmair presented geometric multigrid methods for elliptic problems in the spaces H(curl) and H(div). Their new multigrid components are special smoothers, the grid transfer is canonical since the spaces are nested. Their analysis is much more involved then the multigrid estimates for problems in Sobolev spaces. The difficulty occurs since the operator leads to a regularity pick-up only on a part of the space. They overcome the difficulty by splitting the space due to the discrete Helmholtz decomposition. Heavily exploring the commuting diagram property, they can approximate both components independently.

In our talk we will present a technique to maintain the commuting diagram property during an algebraic coarsening process. The key is to identify the geometric entities node, edge and face, and keep their relationships on the coarser levels. As usual, we can give a two level analysis, and present a couple of numerical results demonstrating the good multigrid behavior. The algorithm is implemented into Stefan Reitzinger's AMG code PEBBLES.