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Volume 12, Issue 4
Error Estimates and Superconvergence of a High-Accuracy Difference Scheme for a Parabolic Inverse Problem with Unknown Boundary Conditions

Liping Zhou, Shi Shu & Haiyuan Yu

DOI: 10.4208/nmtma.OA-2018-0019

Numer. Math. Theor. Meth. Appl., 12 (2019), pp. 1119-1140.

Published online: 2019-06

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  • 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.

  • Keywords

Parabolic inverse problem, unknown boundary condition, finite difference method, discrete Fourier transform, asymptotic optimal order, superconvergence.

  • AMS Subject Headings

65M06, 65M12, 65T50

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

shushi@xtu.edu.cn (Shi Shu)

  • BibTex
  • RIS
  • TXT
@Article{NMTMA-12-1119, author = {Zhou , Liping and Shu , Shi and Yu , Haiyuan}, title = {Error Estimates and Superconvergence of a High-Accuracy Difference Scheme for a Parabolic Inverse Problem with Unknown Boundary Conditions}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2019}, volume = {12}, number = {4}, pages = {1119--1140}, 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.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.OA-2018-0019}, url = {http://global-sci.org/intro/article_detail/nmtma/13217.html} }
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 -

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.

Liping Zhou, Shi Shu & Haiyuan Yu. (2019). Error Estimates and Superconvergence of a High-Accuracy Difference Scheme for a Parabolic Inverse Problem with Unknown Boundary Conditions. Numerical Mathematics: Theory, Methods and Applications. 12 (4). 1119-1140. doi:10.4208/nmtma.OA-2018-0019
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