TY - JOUR
T1 - Error Estimates and Superconvergence of a High-Accuracy Difference Scheme for a Parabolic Inverse Problem with Unknown Boundary Conditions
AU - Zhou , Liping
AU - Shu , Shi
AU - Yu , Haiyuan
JO - Numerical Mathematics: Theory, Methods and Applications
VL - 4
SP - 1119
EP - 1140
PY - 2019
DA - 2019/06
SN - 12
DO - http://doi.org/10.4208/nmtma.OA-2018-0019
UR - https://global-sci.org/intro/article_detail/nmtma/13217.html
KW - Parabolic inverse problem, unknown boundary condition, finite difference method, discrete Fourier transform, asymptotic optimal order, superconvergence.
AB - <p style="text-align: justify;">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&nbsp; and prove that they have superconvergence properties. In the end, numerical experiment demonstrates the theoretical results.</p>