The Performance of the Collocation and Galerkin Methods with Hermite Bi-Cubics

This paper presents a study of the performance of the collocation and Galerkin methods using Hermite bi-cubic basis functions. It is a sequel to the studies of Houstis et al. [6] and Weiser et al. [15]. The two methods have the linear systems solved by direct methods, band Gauss elimination or Cholesky factorization. The problem domain consists of linear, self-adjoint elliptic equations on two-dimensional rectangular domains. The measures of performance are computer time and memory needed to achieve moderate accuracy. The earlier study comparing finite element and finite difference methods observes that collocation uses less computer time than Galerkin. The second study gave detailed operation counts which support this observation, but also gave substantial experimental evidence to the contrary. We use a new implementation of the collocation method by E. N. Houstis which is tailored for rectangular domains (the one used in Houstis et al. [6] was designed for general domains). We use the same Galerkin implementation that Weiser et al. used. We outline the process of comparing the performance of PDE software and discuss the difficulty of reaching definite conclusions. We analyze the question of error measurement and note that the example given in Weiser et al. [15] as a counterexample to the practice of measuring the error at the grid points or knots (as done in Houstis et al. [6]) is a singular problem and also a counterexample to the approach of measuring the error on a fixed set of points. This study strongly supports the hypothesis that (with these implementations of the methods) collocation performs better than Galerkin for computer time and versus error. We observe that a feature of the particular computing environment causes the Galerkin program to use twice as much memory as necessary and a “normal” implementation should be expected to use nearly the same memory as the collocation program.