1. Introduction. In a paper [6], A. A. Samarskii first proposed a "locally one-dimensional" finite difference scheme for the first boundary problem for a parabolic equation where the cross-section of the cylindrical domain involved was arbitrary. He analyzed the scheme in maximum norm and by means of a discrete form of the maximum principle was able to obtain estimates for the order of convergence. These estimates range from 0(t + h) to 0(t + h2) depending on the nature of the cross-section,

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... Hubbard [4]. In [4] a number of locally one-dimensional schemes are proposed and error analyses made in maximum norm which are either 0(t + h) or 0(r + h2) depending on the particular scheme employed. The point of view taken was to define a finite difference analog at each time level whose matrix was a product of tridiagonal matrices. Such an approach has the merit of allowing a more precise analysis of the contribution to the error from "regular interior" points and "irregular interior" points, i.e. those near the boundary. Such an analysis sheds some light on the difficulties involved in formulating 0(r + h ) schemes for general cross-sections. In this paper we adopt the point of view of Samarskii [6] and, using his decomposition of the error, we formulate and analyze a series of economical difference schemes for arbitrary cross-sections whose order of convergence ranges from 0(t + h) to 0(t + h2) depending on the scheme chosen. The techniques employed in the error analysis are related to those used by Bramble and Hubbard [1] and elaborated in later papers. Economical difference schemes were first suggested in 1955 by J. Douglas [2] and Peaceman and Rachford [5] and since then a vast literature has arisen in this area. The reader is referred to the paper of Samarskii [6] and Douglas and Gunn [3] for further references.