Error identities for parabolic initial boundary value problems

S. I. Repin

Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 49, Tome 508 (2021), pp. 147-172

Résumé

The paper is concerned with error identities for a class of parabolic equations. One side of such an identity is a natural measure of the distance between a function in the corresponding energy class and the exact solution of the problem in question. Another side is either directly computable or serves as a source of fully computable error bounds. Particular forms of the identities can be viewed as analogs of the hypercircle identity well known for elliptic problems. It is shown that identities possess an important consistency property. Therefore, the identities and the corresponding error estimates can be used in quantitative analysis of direct and inverse problems associated with parabolic equations. The first part of the paper deals with linear parabolic equations. A class of nonlinear problems is considered in the second part. In particular, this class includes problems, whose spatial parts are presented by the $\alpha$-Laplacian operator.