Low-constant parallel algorithms for finite element simulations using linear octrees

In this article we propose parallel algorithms for the construction of conforming finite-element discretization on linear octrees. Existing octree-based discretizations scale to billions of elements, but the complexity constants can be high. In our approach we use several techniques to minimize overhead: a novel bottom-up tree-construction and 2:1 balance constraint enforcement; a Golomb-Rice encoding for compression by representing the octree and element connectivity as an Uniquely Decodable Code (UDC); overlapping communication and computation; and byte alignment for cache efficiency. The cost of applying the Laplacian is comparable to that of applying it using a direct indexing regular grid discretization with the same number of elements. Our algorithm has scaled up to four billion octants on 4096 processors on a Cray XT3 at the Pittsburgh Supercomputing Center. The overall tree construction time is under a minute in contrast to previous implementations that required several minutes; the evaluation of the discretization of a variable-coefficient Laplacian takes only a few seconds.