Data structures and algorithms for the finite element method on a data parallel supercomputer

This article describes a formulation of the finite element method and its implementation on a data parallel computing system. The Connection Machine® system, CM‐2, has been used as the model architecture. Data structures, storage requirements, communication and parallel arithmetic complexity are analysed in detail for the cases when a processor represents an unassembled finite element and when a processor is assigned to an unassembled nodal point. Data parallel algorithms for the grid generation, the evaluation of the elemental stiffness matrices and for the iterative solution of the linear system are presented. The algorithm for evaluating the elemental stiffness matrices computes the matrix elements concurrently without communication. This concurrency is in addition to the inherent parallelism present among different finite elements. A conjugate gradient solver with diagonal pre‐conditioner is used for the solution of the resulting linear system. Results from an implementation of the three‐dimensional finite element method based on Lagrange elements are reported. For single‐precision floating‐point operations, the measured peak performance is approximately 2·4 G flops s−1 for evaluating the elemental stiffness matrices and approximately 850 M flops s−1 for the conjugate gradient solver. On a Connection Machine system with 16K physical processors, the time per conjugate gradient iteration for an application with 400 000 degrees of freedom is approximately 0·13 s for double‐precision floating‐point operations.