An Extension of the Alternating Direction Galerkin Method to More General Geometries

Abstract

In this paper an extension of the alternating direction Galerkin method to rectangular polygons is presented. Only Dirichlet problems with tensor product basis of piecewise linear Hermite polynomials are considered. The approach proposed is based on imbedding the rectangular polygon into a rectangle, and then using a result due to F. Stenger [5] to provide the solution of the associated algebraic problem in terms of a sequence of simpler algebraic problems. This approach has strong similarities with that proposed by J. Dendy and G. Fairweather [SIAM J. Numer. Anal., 12 (1975), pp. 144–163]; however, it is conceptually different. Furthermore, the solution is given in an algorithmic manner that is well suited to computer implementation, and it is also more general for the class of problems considered.