Alternating-Direction Galerkin Methods for Parabolic and Hyperbolic Problems on Rectangular Polygons

Douglas and Dupont have formulated several alternating-direction Galerkin schemes on rectangular regions. In this paper it is shown that for certain time-dependent problems on rectangular polygons these schemes can be implemented with no more work than in the rectangular case; in fact the number of arithmetic operations required to complete one time step is O(N), where N is the number of unknowns at each time step. The results extend to systems of nonlinear parabolic equations and to problems on unions of rectangular parallelepipeds in Rn.