On the Numerical Computation of Parabolic Problems for Preceding Times

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. 1. Introduction. Beginning with Hadamard, who drew attention to such prob- lems, many analysts have been attracted to the study of improperly posed problems in mathematical physics. A recent survey by Payne in (22) lists over fifty references. Further references are to be found in (15), (16), (14), (3), (10), and (1). The two best known examples of ill-posed problems are the Cauchy problem for Laplace's equation and the Cauchy problem for the backward heat equation. Some remarks concerning practical interest in such questions can be found in (7, p. 231), (15), (27) and (28). 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

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