Development, evaluation and selection of methods for elliptic partial differential equations

We present a framework within which to evaluate and compare computational methods to solve elliptic partial differential equations. We then report on the results of comparisons of some classical methods as well as a new one presented here. Our main motivation is the belief that the standard finite difference methods are almost always inferior for solving elliptic problems and our results are strong evidence that this is true. The superior methods are higher order (fourth or more instead of second) and we describe a new collocation finite element method which we believe is more efficient and flexible than the other well known methods, e.g., fourth order finite differences, fourth order finite element methods of Galerkin, Rayleigh-Ritz or least squares type. Our comparisons are in the context of the relatively complicated problems that arise in realistic applications. Our conclusion does not hold for simple model problems (e.g., Laplaces equation on a rectangle) where very specialized methods are superior to the generally applicable methods that we consider. The accurate and relatively simple treatment of boundary conditions involving curves and derivations is a feature of our collocation method.