Using Matrix-free, Multigrid Methods in ASPECT for Large Scale Computations

Solving discrete finite element systems rely heavily on the computation of matrix-vector products with sparse matrices. These matrix-vector products tend to dominate the total amount of work required for the average finite element program, and, as these matrices can be quite large and do not fit into the cache of a modern machine, accessing the data from RAM has become a major bottleneck in finite element computations. Instead of computing the entries of and storing a system matrix, matrix-free methods define an operation on a vector as a loop over the cells in the domain, applying small, dense matrices defined locally on the cell, and summing the results over the entire domain. In this way, one replicates the action of a matrix while never actually storing a matrix. These methods show significant gains in the time of a matrix-vector product as compared to traditional matrix-based methods when using degree 2 and higher finite elements [1]. [full article]

Reference

[1] M. Kronbichler and K. Kormann. A generic interface for parallel cell-based finite element operator application. Computers & Fluids, 63:135-147, 2012.

[2] T.C. Clevenger, T.Heister, G. Kanschat, and M. Kronbichler. A flexible, parallel, adaptive geometric multigrid method for fem. submitted.

Contributed by Thomas Clevenger