Finite Element Integration on GPUs

We present a novel finite element integration method for low order elements on GPUs. We achieve more than 100GF for element integration on first order discretizations of both the Laplacian and Elasticity operators.Comment: 16 pages, 3 figure

Topological Optimization of the Evaluation of Finite Element Matrices

We present a topological framework for finding low-flop algorithms for evaluating element stiffness matrices associated with multilinear forms for finite element methods posed over straight-sided affine domains. This framework relies on phrasing the computation on each element as the contraction of each collection of reference element tensors with an element-specific geometric tensor. We then present a new concept of complexity-reducing relations that serve as distance relations between these reference element tensors. This notion sets up a graph-theoretic context in which we may find an optimized algorithm by computing a minimum spanning tree. We present experimental results for some common multilinear forms showing significant reductions in operation count and also discuss some efficient algorithms for building the graph we use for the optimization

Common and Unusual Finite Elements

This chapter provides a glimpse of the considerable range of finite elements in the literature. Many of the elements presented here are implemented as part of the FEniCS project already; some are future work. The universe of finite elements extends far beyond what we consider here. In particular, we consider only simplicial, polynomial-based elements. We thus bypass elements defined on quadrilaterals and hexahedra, composite and macro-element techniques, as well as XFEM-type methods. Even among polynomial-based elements on simplices, the list of elements can be extended. Nonetheless, this chapter presents a comprehensive collection of some the most common, and some more unusual, finite elements

Discrete Optimization of Finite Element Matrix Evaluation

The tensor contraction structure for the computation of the element tensor AT obtained in Chapter 8,enables not only the construction of a compiler for variational forms,but an optimizing compiler.For typical variational forms,the reference tensor A0 has significant structure that allows the element tensor AT to be computed on an arbitrary cell T at a lower computational cost