Smoothed Aggregation for Higher-Order

Our higher-order research has focused on DG discretizations of the Poisson problem. In addition to the difficulties posed by the discretization, as the polynomial order of the finite element basis increases, a number of difficulties arise for classic multigrid techniques. The discrete operator no longer forms an M-matrix, which affects basic multigrid components such as strength-of-connection. Also, standard point-wise relaxation and classic interpolation methods, such as standard prolongation smoothing, degrade. Our current goal is to develop an effective multigrid preconditioner for higher-order DG discretizations of the Poisson problem. In the future, we look to extend our methods to more challenging problems.

Our work so far has focused on extending classic smoothed aggregation techniques, i.e., aggregation and prolongation smoothing, to address these difficulties. For prolongation smoothing, Krylov methods operating in a constraint space are used. For an example aggregation, see the below picture. This approach has yielded promising results. In the future, we plan to build on previous work which has utilized lower-order discretizations, which are known to precondition higher-order problems well.

In the below image, a sample aggregation of higher-order DG elements is given. Aggregates are denoted by the blue polygons and lines. Nodes are denoted by the small squares. The elements of the discontinuous mesh are depicted in yellow, and have been visually scaled to 85% of their original size, so that their boundaries don't overlap.

Last updated September, 2009