Multigrid Preconditioning for Matrix-free Operators

Thomas Anderson, California Institute of Technology

We present several novel methods to solve high-order finite element discretizations in matrix-free form. One means to increase the accuracy of a finite element discretization is to increase the polynomial order, an approach that can be particularly advantageous on emerging computer architectures that exploit a high degree of parallelism. The high-order discretization leads to dense submatrices whose memory access patterns can be known in advance, and in some settings high-speed tensor contractions can be exploited. The use of higher-order discretizations also makes it progressively more expensive to explicitly assemble the associated element matrices and therefore correspondingly more attractive to apply the matrix without storing its entries. Though this "matrix free" setting is attractive from one perspective, it makes preconditioning and especially algebraic preconditioning a challenge. Here we consider ways to assemble sparser versions of such an operator that maintains its essential features. The sparsification strategies may involve reducing the number of points per element in each dimension or relying on a new low-order discretization on the same physical grid. Regardless of the approach, once such a sparsification is assembled, an algebraic multigrid preconditioner can be developed. We find that this AMG preconditioner can be effective as a preconditioner for the original matrix-free operator and present scaling data showcasing the method's performance as a solver.

Abstract Author(s): Thomas Anderson, Andrew Barker, Tzanio Kolev