Evaluation of numerical methods for elliptic partial differential equations

We systematically evaluate four methods for solving two-dimensional, linear elliptic partial differential equations on general domains. The four methods are: standard finite differences; collocation, Galerkin, and least squares using Hermite cubic piecewise polynomials. The collocation method is new in that it applies to general curved domains and we describe this aspect in detail. Our test set of 17 problems ranges from simple to moderately complex. The principal conclusion is that collocation is the most efficient method for general use. Standard finite differences is sometimes more efficient for very crude accuracy (where efficiency is not important anyway) but it is also sometimes enormously less efficient even for very modest accuracy. The accuracy of the Galerkin and least-squares methods is sometimes better than collocation, but the extra cost always negates this advantage for our problems.