Recently, several investigators have discussed both analyticallS7 and iterative approaches to determine optimal (minimum potential energy) grid co-ordinates for a given differential equation, boundary conditions and topological finite element arrangement of specific shape functions. In general, it is probably fair to say that, except for formulations involving very few degrees of freedom, the benefits to be gained from the best solution of a given connectivity, as compared with that obtained from a mesh which would routinely be generated by an experienced practitioner or mesh-generating program, are not worth the computational costs entailed in iteratively computing the changing grid co-ordinates of the coarser mesh. The objective of the researches cited has been to develop guidelines for construction of near-optimal meshes for analysis of the more complicated problems encountered in engineering practice. Of course, if explicit formulae for the optimal grid of a given connectivity can be obtained a prioril then this grid should be used.
One important area where computational and analytical methods for determining optimal meshes of standard finite elements would be of value is in the vicinity of a sing~larity,~ although it is also possible to incorporate the singularity into the space of finite element trial functions.8 Fried and Yang7 have shown that an appropriate grading of a finite element mesh in the vicinity of a singularity can lead to the full order of convergence (in terms of number of degrees of freedom used) which would be expected using equal-sized elements of the same order polynomial shape functions in a problem which was free of singularities. This full order of convergence for graded meshes is in marked contrast to the results of Tong and Pian' who showed that, for equal-sized elements, the order of convergence of the finite element solution is independent of the order of the polynomial shape functions and depends only on the order of the singularity. Fried and Yang7 suggested an approximate scheme for 'best' grading of finite elements at a singularity which depended on the order of the singularity, the order of the polynomial shape functions and the dimension of the space in which the singularity is embedded. In addition, they offered numerical solutions based on this approximate best-grading which obtained 'nearly' the full order of convergence expected.
It is the purpose of this communication to use some of the analytical techniques for determining precisely optimal grids in a particular one-dimensional equation having a singularity, and examine the convergence properties of regular, optimal and near-optimal meshes.
Example problem
Consider the differential equation (A(x)u'(x))' = 0 on [0,1] with boundary conditions u(0) = 0, u'(1) = 1. Here the prime denotes differentiation with respect to spatial variable x. The