Introduction

The finite element (FE) method (e.g., Johnson, 1987) is well suited to problems with irregular computational domains and problems that have solutions whose smoothness varies greatly from one part of the domain to another. It has these properties because the approximating solution is expressed as a superposition of basis functions with local support. Irregular domains can be accommodated because the domain is divided into elements that need not be laid out on a regular grid. A higher density of elements can be used where the solution is complex and a lower density can be used where the solution is smooth. This is in contrast to the finite difference method that uses a regular grid with the same spacing throughout the computational domain.

The FE solution is expressed as a function of the values at nodal points. The nodal values are obtained by solving a sparse set of linear equations. The structure of the sparse, symmetric operator is controlled by the layout of the elements and nodes. The dimension of the operator is the number of nodes in the problem. A row of the operator corresponding to one node will have non-zero entries in the columns corresponding to nodes that share a common element with that node. In this paper I will show that the regularity of the layout of the elements is an important factor in the efficiency of the method on a CM.

I illustrate the method with a model problem. In the model problem I use rectangular elements with bilinear basis functions. The solution of this problem on a CM requires a different approach to that used for vector processors. I am able to make efficient use of the CM by using the conjugate gradient method (e.g. Golub & Van Loan, 1989). The speed of the solution appears to be controlled by the speed at which the CM can execute a dot product of two arrays.

If the elements are not laid out on a regularly connected mesh the operator is more expensive to apply on a CM. However, more spatially complex problems can be solved, as this is the type of problem that the FE method is designed to solve. The major challenge in implementing FE methods on the Connection Machine is to write efficient algorithms when the nodes are not regularly connected.