Some efficient and accurate direct methods are developed for solving certain elliptic partial difference equations over a rectangle with Dirlchlet. Neumann or periodic boundary conditions. Generalizations to higher dimensions and to L-shaped regions are included.

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Abstract. We develop and analyze a general procedure for computing selfadjoint parabolic problems backwards in time, given an a priori bound on the solutions. The method is applicable to mixed problems with variable coefficients which may depend on time. We obtain error bounds which are naturally related to certain convexity inequalities in parabolic equations. In the time-dependent case, our difference scheme discerns three classes of problems. In the most severe case, we recover a convexity result of Agmon and Nirenberg. We illustrate the method with a numerical experiment. From the viewpoint of numerical analysis, this ill-posedness manifests itself in the most serious way. We have discontinuous dependence on the data. Consequently [24, p. 59], every finite-difference scheme consistent with such a problem, and which is implemented as a marching process, is necessarily unstable. On the other hand, as was observed by John in [12] and Pucci in [23], continuous dependence can often be restored by requiring the solutions to satisfy a suitable constraint. Typically, one asks for nonnegative solutions or for solutions which satisfy an a priori bound, obtainable from physical considerations. The problem then is one of incorporating the

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