Error Estimates and Superconvergence of a High-Accuracy Difference Scheme for a Parabolic Inverse Problem with Unknown Boundary Conditions

Abstract

In this work, we firstly construct an implicit Euler difference scheme for a one-dimensional parabolic inverse problem with an unknown time-dependent function in the boundary conditions. Then we initially prove that this scheme can reach the asymptotic optimal error estimate in the maximum norm. Next, we present some approximation formulas for the solution derivative and the unknown boundary function  and prove that they have superconvergence properties. In the end, numerical experiment demonstrates the theoretical results.