Some aspects of parallel implementation of the finite-element method on message passing architectures

We discuss some aspects of implementing the finite-element method on parallel computers with local memory and message passing. In particular, we compare the costs of using high-order and low-order elements and of direct and iterative solvers for solving the linear systems that occur. Our model of parallel computation is a two-dimensional grid of processors chosen to be similar in shape to the underlying grid. Our main conclusions are that use of high-order methods is an effective way to achieve high accuracy for some problems, on both serial and parallel computers, and that such methods provide a natural way to achieve efficiency in parallel implementations. In addition, we show that sparse direct solvers generalize naturally to methods based on high-order elements, and that direct solvers are adequate for two-dimensional problems, especially for multiple load vectors.