Parallel Implementation of the $hp$-Version of the Finite Element Method on a Shared-Memory Architecture

Abstract

The costs incurred by an implementation of the $hp$-version of the finite element for solving two-dimensional elliptic partial differential equations on a shared-memory parallel computer are studied. For a collection of benchmark problems, the costs in CPU time of various individual subtasks performed by the finite element solver are systematically examined, including construction of local stiffness matrices, elimination of unknowns associated with element interiors, and global solution on element interfaces by a preconditioned conjugate gradient method. General observations are that the costs of the “naturally” parallel computations associated with local elements are significantly higher than any global computations, so that the latter do not represent a significant bottleneck to parallel efficiency. However, memory conflicts place some limitations on the sizes or number of local problems that can be handled efficiently in parallel.