In this paper, we consider the optimal source control of a 2-dimensional steady-state thermistor. The problem is described by a system of two nonlinear elliptic partial differential equations with appropriate boundary conditions which model the coupling of the thermistor to its surroundings. The heat source is Joule heat due to variable resistance. The problem is a source optimal control problem that controls the source term necessary to approximate the temperature to a proper target function. First, we derive the optimality condition of the problem. Based on setting the approximation problem of a given control problem in a first order polynomial finite element function space and deriving the optimality condition of the approximation problem, we evaluated a priori error between the optimal control, the optimal state, the conjugate state and its finite element approximation functions. Then, we evaluate the upper bound of a posteriori error estimates that are currently available for error estimation. For a posteriori error estimates, it is necessary to find the convergence of the error indicator. In this paper, we prove the convergence of a posteriori error indicator by obtaining a lower bound estimate of a posteriori error and finding that the total variance error goes to zero. And, we propose a gradient algorithm to find the optimal control and provide a condition for this algorithm to converge. The validity is also demonstrated by adaptive numerical simulations with a detailed problem. The computational results are obtained on three adaptive meshes and the graphs of the finite element solutions are presented.
| Published in | International Journal of Theoretical and Applied Mathematics (Volume 11, Issue 1) |
| DOI | 10.11648/j.ijtam.20251101.11 |
| Page(s) | 1-17 |
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This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. |
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Copyright © The Author(s), 2025. Published by Science Publishing Group |
Quasi-linear Elliptic Equations, Source Control, A Posteriori Error Estimates
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APA Style
Kim, C., Ri, J. (2025). A Priori and a Posteriori Error Estimates of Finite Element Method for Source Control Problems Governed by a System of Quasi-linear Elliptic Equations. International Journal of Theoretical and Applied Mathematics, 11(1), 1-17. https://doi.org/10.11648/j.ijtam.20251101.11
ACS Style
Kim, C.; Ri, J. A Priori and a Posteriori Error Estimates of Finite Element Method for Source Control Problems Governed by a System of Quasi-linear Elliptic Equations. Int. J. Theor. Appl. Math. 2025, 11(1), 1-17. doi: 10.11648/j.ijtam.20251101.11
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Download@article{10.11648/j.ijtam.20251101.11,
author = {Changil Kim and Jayong Ri},
title = {A Priori and a Posteriori Error Estimates of Finite Element Method for Source Control Problems Governed by a System of Quasi-linear Elliptic Equations},
journal = {International Journal of Theoretical and Applied Mathematics},
volume = {11},
number = {1},
pages = {1-17},
doi = {10.11648/j.ijtam.20251101.11},
url = {https://doi.org/10.11648/j.ijtam.20251101.11},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijtam.20251101.11},
abstract = {In this paper, we consider the optimal source control of a 2-dimensional steady-state thermistor. The problem is described by a system of two nonlinear elliptic partial differential equations with appropriate boundary conditions which model the coupling of the thermistor to its surroundings. The heat source is Joule heat due to variable resistance. The problem is a source optimal control problem that controls the source term necessary to approximate the temperature to a proper target function. First, we derive the optimality condition of the problem. Based on setting the approximation problem of a given control problem in a first order polynomial finite element function space and deriving the optimality condition of the approximation problem, we evaluated a priori error between the optimal control, the optimal state, the conjugate state and its finite element approximation functions. Then, we evaluate the upper bound of a posteriori error estimates that are currently available for error estimation. For a posteriori error estimates, it is necessary to find the convergence of the error indicator. In this paper, we prove the convergence of a posteriori error indicator by obtaining a lower bound estimate of a posteriori error and finding that the total variance error goes to zero. And, we propose a gradient algorithm to find the optimal control and provide a condition for this algorithm to converge. The validity is also demonstrated by adaptive numerical simulations with a detailed problem. The computational results are obtained on three adaptive meshes and the graphs of the finite element solutions are presented.},
year = {2025}
}
TY - JOUR T1 - A Priori and a Posteriori Error Estimates of Finite Element Method for Source Control Problems Governed by a System of Quasi-linear Elliptic Equations AU - Changil Kim AU - Jayong Ri Y1 - 2025/06/21 PY - 2025 N1 - https://doi.org/10.11648/j.ijtam.20251101.11 DO - 10.11648/j.ijtam.20251101.11 T2 - International Journal of Theoretical and Applied Mathematics JF - International Journal of Theoretical and Applied Mathematics JO - International Journal of Theoretical and Applied Mathematics SP - 1 EP - 17 PB - Science Publishing Group SN - 2575-5080 UR - https://doi.org/10.11648/j.ijtam.20251101.11 AB - In this paper, we consider the optimal source control of a 2-dimensional steady-state thermistor. The problem is described by a system of two nonlinear elliptic partial differential equations with appropriate boundary conditions which model the coupling of the thermistor to its surroundings. The heat source is Joule heat due to variable resistance. The problem is a source optimal control problem that controls the source term necessary to approximate the temperature to a proper target function. First, we derive the optimality condition of the problem. Based on setting the approximation problem of a given control problem in a first order polynomial finite element function space and deriving the optimality condition of the approximation problem, we evaluated a priori error between the optimal control, the optimal state, the conjugate state and its finite element approximation functions. Then, we evaluate the upper bound of a posteriori error estimates that are currently available for error estimation. For a posteriori error estimates, it is necessary to find the convergence of the error indicator. In this paper, we prove the convergence of a posteriori error indicator by obtaining a lower bound estimate of a posteriori error and finding that the total variance error goes to zero. And, we propose a gradient algorithm to find the optimal control and provide a condition for this algorithm to converge. The validity is also demonstrated by adaptive numerical simulations with a detailed problem. The computational results are obtained on three adaptive meshes and the graphs of the finite element solutions are presented. VL - 11 IS - 1 ER -
Department of Mathematics, University of Sciences, Pyongyang, DPR Korea
Department of Mathematics, University of Sciences, Pyongyang, DPR Korea
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