A numerical method for backward parabolic problems with non-selfadjoint elliptic operators

Abstract

A method of solution of backward parabolic problems with non-selfadjoint elliptic operators is presented. The method employs a quasisolution approach and is based on the separation of the problem into a sequence of well-posed forward problems on the entire mesh and an ill-posed system of algebraic equations on a coarser submesh. For the corresponding forward problem the continuous dependence of the solution on the initial profile is proved. From this result a stability estimate on the final time T is obtained. The estimate shows a decrease in stability of the forward (hence, the backward) problem, as the final time T is increased. Using the stability result the existence of a quasisolution of the backward problem is proved. For the solution of the intermediate non-selfadjoint forward problems a modified alternating-direction finite difference scheme is presented. The ill-conditioned system of algebraic equations is solved by using truncated singular value decomposition. The effectiveness of the method is demonstrated on a numerical test problem with exact and noisy data.