A Collocation-Galerkin Method for Poisson's Equation on Rectangular Regions

A collocation-Galerkin method is defined for Poisson's equation on the unit 2 square, using tensor products of continuous piecewise polynomials. Optimal L and Hq orders of convergence are established. This procedure requires fewer quadratures than the corresponding Galerkin procedure. Introduction. The collocation-Galerkin method was first introduced by Diaz [2], [3] for the two-point boundary value problems and optimal L2 -rates of convergence were established for a particular choice of the collocation points; namely, the affine images of the roots of a Jacobi polynomial. In [6] Wheeler derived optimal V-estimates and applied this method to a one space dimensional parabolic problem. In [4] Dunn and Wheeler analyzed some collocation-//-1 -Galerkin methods and established optimal Vestimates for any choice of the collocation points. Archer and Diaz [1] have applied similar ideas to a one-dimensional first order hyperbolic problem and derived optimal ¿2-estimates. Here, a collocation-Galerkin method is defined for Poisson's equation on the unit square, using tensor products of continuous piecewise polynomials and the collocation points are based on the roots of a Jacobi polynomial. Optimal L2and//¿-estimates are established. On the basis of computational complexity the collocation-Galerkin method is intermediate between the Galerkin method and the collocation method. It has an advantage over the Galerkin procedure for the same space in that the integrals involve the product of the approximate solution and a piecewise linear function, thus the integrals are simpler and, of course, there are fewer of them. Also, the continuity conditions on the approximate solution are weaker than those required of the collocation approximation defined by Prenter and Russell [5]. In the following section, the collocation-Galerkin method is defined and existence and uniqueness are shown using some semidiscrete bilinear forms. In the last section, the error analysis is presented. The analysis consists of reducing the problem to some one-dimensional problems for which the results of [3] can be applied. The Collocation-Galerkin Method. Consider the boundary value problem A77 = /, on Í2, u = 0, on oil, where Í2 = (0, 1) x (0, 1). We shall assume that there exists a unique u and that it is sufficiently smooth. Received March 8, 1976; revised March 24, 1978. AMS (MOS) subject classifications (1970). Primary 65N35, 65N30. © 1979 American Mathematical Society 0025-571 8/79/0000-0005/S03.00 77 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use