A Posteriori Error Estimates by FEM for Source Control Problems Governed by a System of Semi-Linear Convection-Diffusion Equations

ChangIl Kim; JaYong Ri; Song Jun Kim

doi:doi:10.11648/j.ijimse.20251002.11

Research Article |

| Peer-Reviewed

A Posteriori Error Estimates by FEM for Source Control Problems Governed by a System of Semi-Linear Convection-Diffusion Equations

ChangIl Kim^*, JaYong Ri, Song Jun Kim

Published in International Journal of Industrial and Manufacturing Systems Engineering (Volume 10, Issue 2)

Received: 24 January 2025 Accepted: 7 August 2025 Published: 23 September 2025

Views: Downloads:

Download PDF

Share This Article

Twitter
Linked In
Facebook

Abstract

In this paper, we consider the optimal source control problem of a system of 2-dimensional semi-linear steady convection-diffusion equations. The problem is modelized from temperature and consistency distribution in the gasification processes, so it is described by 2 non-linear elliptic partial differential equations with Dirichlet boundary condition. The problem is a optimal source control problem that controls the source term necessary to approximate the temperature to a proper target function. First, we derived the optimal condition. 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 by using optimal condition of original and approximate problem. And we also evaluated the upper estimate of a posteriori error by finite element method (FEM). We proved the convergence to 0 of a posteriori error indicator (term of the right side of inequality) when division diameter converges to 0. For this, we acquired the lower bound estimation of a posteriori error and proved that a priori error and total variance error converges to 0 when division diameter converges to 0, so that we proved the convergence problem of a posteriori error indicator.

Published in	International Journal of Industrial and Manufacturing Systems Engineering (Volume 10, Issue 2)
DOI	10.11648/j.ijimse.20251002.11
Page(s)	20-35
Creative Commons	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.
Copyright	Copyright © The Author(s), 2025. Published by Science Publishing Group

Keywords

System of Semi-Linear Convection-Diffusion Equations, Source Control, A Posteriori Error Estimates

References

[1]	Bei Zhang et al, A posteriori error analysis of nonconforming finite element methods for convection-diffusion problems, Journal of Computational and Applied Mathematics, 321, 416-426 (2017).
[2]	R. Verfurth, A posteriori error estimates for non-stationary nonlinear convection-diffusion equation, Calcolo, 55, 1-18 (2018).
[3]	Chuanjun Chen et al., A posteriori error Estimates of two grid finite volume element method for non-linear elliptic problem, Computers and Mathematics with Applications, 75(2018) 1756-1766.
[4]	Xingyang Ye, Chanju Xu, A Posteriori error estimates for the fractional optimal control problems, Journal of Inequalities and Applications, 141(2015), 1-13.
[5]	Lin Li et al., A posteriori error Estimates of spectral method for non-linear parabolic optimal control problem, Journal of inequalities and applications, 138(2018) 1-23.
[6]	E. Casas et al., Error Estimates for the numerical approximation of Dirichlet boundary control for semilinear elliptic equations, SIAM J. Control Optim., 45(2006) 1586-1611.
[7]	D. Y. Shi, H. J. Yang, Superconvergence analysis of finite element method for time-fractional thermistor problem, Appl. Math. Comput., 323 (2018) 31–42.
[8]	D. Y. Shi, H. J. Yang, Superconvergence analysis of nonconforming FEM fornonlinear time-dependent thermistor problem, Appl. Math. and Compu., 347 (2019) 210–224.
[9]	Y. Chen, L. Chen, X. Zhang, Two-grid method for nonlinear parabolic equations by expande mixed finite element methods, Numer. Methods Part. Diff. Equ., 29(2013) 1238-1256.
[10]	Meyer C., Error estimates for the finite element approximation of an elliptic control problem with pointwise state and control constraints, Control and Cybern., 37(1), 51-83 (2008).
[11]	Wollner W., A posteriori error estimates for a finite element distretization of Interior point methods for an elliptic optimization problem with state constraints, Numer. Math. Vol. 120, No. 4, 133-159 (2012).
[12]	Rosch, D. Wachsmuth, A posteriori error estimates for optimal control problems with state and control constraints, Numerische Mathematik, Vol. 120, No. 4, 733-762 (2012).
[13]	Benedix, B. Vexler, A posteriori error estimation and adaptivity for elliptic optimal control problems with state constraints, Comput. Optim. Appl., 44(1), 3-25 (2009).
[14]	Dib S., Girault V., Hecht F. and Sayah T., A posteriori error estimates for Darcy’s problem coupled with the heat equation, ESAIM Mathematical Modelling and Numerical Analysis, https://doi.org/10.1051/m2an/2019049 (2019).
[15]	Allenes, E. Otarola, R. Rankin. A posteriori error estimation for a PDE constrained optimization problem involving the generalized Oceen equations, SIAM J. Sci. Comput., Vol. 40, No. 4, A2200-A2233, 2018.
[16]	Natalia Kopteva. Error analysis of the L1 method on graded and uniform meshes for a fractional-derivative problem in two and three dimensions. Math. Comp., 88(319): 2135–2155, 2019.
[17]	Xiangcheng Zheng and Hong Wang. Optimal-order error estimates finit element approximations to variable-order time-fractional diffusion equations without regularity assumptions of the true solutions. IMA J. Numer. Anal., 41(2): 1522–1545, 2021.
[18]	Natalia Kopteva. Pointwise-in-time a posteriori error control for time-fractional parabolic equations. Appl. Math. Lett., 123: Paper No. 107515, 8, 2022.

Cite This Article

Plain Text BibTeX RIS

APA Style

Kim, C., Ri, J., Kim, S. J. (2025). A Posteriori Error Estimates by FEM for Source Control Problems Governed by a System of Semi-Linear Convection-Diffusion Equations. International Journal of Industrial and Manufacturing Systems Engineering, 10(2), 20-35. https://doi.org/10.11648/j.ijimse.20251002.11

Copy | Download

ACS Style

Kim, C.; Ri, J.; Kim, S. J. A Posteriori Error Estimates by FEM for Source Control Problems Governed by a System of Semi-Linear Convection-Diffusion Equations. Int. J. Ind. Manuf. Syst. Eng. 2025, 10(2), 20-35. doi: 10.11648/j.ijimse.20251002.11

Copy | Download

AMA Style

Kim C, Ri J, Kim SJ. A Posteriori Error Estimates by FEM for Source Control Problems Governed by a System of Semi-Linear Convection-Diffusion Equations. Int J Ind Manuf Syst Eng. 2025;10(2):20-35. doi: 10.11648/j.ijimse.20251002.11

Copy | Download

@article{10.11648/j.ijimse.20251002.11,
  author = {ChangIl Kim and JaYong Ri and Song Jun Kim},
  title = {A Posteriori Error Estimates by FEM for Source Control Problems Governed by a System of Semi-Linear Convection-Diffusion Equations
},
  journal = {International Journal of Industrial and Manufacturing Systems Engineering},
  volume = {10},
  number = {2},
  pages = {20-35},
  doi = {10.11648/j.ijimse.20251002.11},
  url = {https://doi.org/10.11648/j.ijimse.20251002.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijimse.20251002.11},
  abstract = {In this paper, we consider the optimal source control problem of a system of 2-dimensional semi-linear steady convection-diffusion equations. The problem is modelized from temperature and consistency distribution in the gasification processes, so it is described by 2 non-linear elliptic partial differential equations with Dirichlet boundary condition. The problem is a optimal source control problem that controls the source term necessary to approximate the temperature to a proper target function. First, we derived the optimal condition. 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 by using optimal condition of original and approximate problem. And we also evaluated the upper estimate of a posteriori error by finite element method (FEM). We proved the convergence to 0 of a posteriori error indicator (term of the right side of inequality) when division diameter converges to 0. For this, we acquired the lower bound estimation of a posteriori error and proved that a priori error and total variance error converges to 0 when division diameter converges to 0, so that we proved the convergence problem of a posteriori error indicator.
},
 year = {2025}
}

Copy | Download

TY  - JOUR
T1  - A Posteriori Error Estimates by FEM for Source Control Problems Governed by a System of Semi-Linear Convection-Diffusion Equations

AU  - ChangIl Kim
AU  - JaYong Ri
AU  - Song Jun Kim
Y1  - 2025/09/23
PY  - 2025
N1  - https://doi.org/10.11648/j.ijimse.20251002.11
DO  - 10.11648/j.ijimse.20251002.11
T2  - International Journal of Industrial and Manufacturing Systems Engineering
JF  - International Journal of Industrial and Manufacturing Systems Engineering
JO  - International Journal of Industrial and Manufacturing Systems Engineering
SP  - 20
EP  - 35
PB  - Science Publishing Group
SN  - 2575-3142
UR  - https://doi.org/10.11648/j.ijimse.20251002.11
AB  - In this paper, we consider the optimal source control problem of a system of 2-dimensional semi-linear steady convection-diffusion equations. The problem is modelized from temperature and consistency distribution in the gasification processes, so it is described by 2 non-linear elliptic partial differential equations with Dirichlet boundary condition. The problem is a optimal source control problem that controls the source term necessary to approximate the temperature to a proper target function. First, we derived the optimal condition. 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 by using optimal condition of original and approximate problem. And we also evaluated the upper estimate of a posteriori error by finite element method (FEM). We proved the convergence to 0 of a posteriori error indicator (term of the right side of inequality) when division diameter converges to 0. For this, we acquired the lower bound estimation of a posteriori error and proved that a priori error and total variance error converges to 0 when division diameter converges to 0, so that we proved the convergence problem of a posteriori error indicator.

VL  - 10
IS  - 2
ER  -

Copy | Download

Author Information

ChangIl Kim

Department of Mathematics, University of Sciences, Pyongyang, DPR Korea

Contact Email
JaYong Ri

Department of Mathematics, University of Sciences, Pyongyang, DPR Korea

Contact Email
Song Jun Kim

Department of Mathematics, Pyongsong University of Education, Pyongsong, DPR Korea

Contact Email

Download PDF

Submit an Article

Sections

Plain Text BibTeX RIS

APA Style

Kim, C., Ri, J., Kim, S. J. (2025). A Posteriori Error Estimates by FEM for Source Control Problems Governed by a System of Semi-Linear Convection-Diffusion Equations. International Journal of Industrial and Manufacturing Systems Engineering, 10(2), 20-35. https://doi.org/10.11648/j.ijimse.20251002.11

Copy | Download

ACS Style

Kim, C.; Ri, J.; Kim, S. J. A Posteriori Error Estimates by FEM for Source Control Problems Governed by a System of Semi-Linear Convection-Diffusion Equations. Int. J. Ind. Manuf. Syst. Eng. 2025, 10(2), 20-35. doi: 10.11648/j.ijimse.20251002.11

Copy | Download

AMA Style

Kim C, Ri J, Kim SJ. A Posteriori Error Estimates by FEM for Source Control Problems Governed by a System of Semi-Linear Convection-Diffusion Equations. Int J Ind Manuf Syst Eng. 2025;10(2):20-35. doi: 10.11648/j.ijimse.20251002.11

Copy | Download

@article{10.11648/j.ijimse.20251002.11,
  author = {ChangIl Kim and JaYong Ri and Song Jun Kim},
  title = {A Posteriori Error Estimates by FEM for Source Control Problems Governed by a System of Semi-Linear Convection-Diffusion Equations
},
  journal = {International Journal of Industrial and Manufacturing Systems Engineering},
  volume = {10},
  number = {2},
  pages = {20-35},
  doi = {10.11648/j.ijimse.20251002.11},
  url = {https://doi.org/10.11648/j.ijimse.20251002.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijimse.20251002.11},
  abstract = {In this paper, we consider the optimal source control problem of a system of 2-dimensional semi-linear steady convection-diffusion equations. The problem is modelized from temperature and consistency distribution in the gasification processes, so it is described by 2 non-linear elliptic partial differential equations with Dirichlet boundary condition. The problem is a optimal source control problem that controls the source term necessary to approximate the temperature to a proper target function. First, we derived the optimal condition. 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 by using optimal condition of original and approximate problem. And we also evaluated the upper estimate of a posteriori error by finite element method (FEM). We proved the convergence to 0 of a posteriori error indicator (term of the right side of inequality) when division diameter converges to 0. For this, we acquired the lower bound estimation of a posteriori error and proved that a priori error and total variance error converges to 0 when division diameter converges to 0, so that we proved the convergence problem of a posteriori error indicator.
},
 year = {2025}
}

Copy | Download

TY  - JOUR
T1  - A Posteriori Error Estimates by FEM for Source Control Problems Governed by a System of Semi-Linear Convection-Diffusion Equations

AU  - ChangIl Kim
AU  - JaYong Ri
AU  - Song Jun Kim
Y1  - 2025/09/23
PY  - 2025
N1  - https://doi.org/10.11648/j.ijimse.20251002.11
DO  - 10.11648/j.ijimse.20251002.11
T2  - International Journal of Industrial and Manufacturing Systems Engineering
JF  - International Journal of Industrial and Manufacturing Systems Engineering
JO  - International Journal of Industrial and Manufacturing Systems Engineering
SP  - 20
EP  - 35
PB  - Science Publishing Group
SN  - 2575-3142
UR  - https://doi.org/10.11648/j.ijimse.20251002.11
AB  - In this paper, we consider the optimal source control problem of a system of 2-dimensional semi-linear steady convection-diffusion equations. The problem is modelized from temperature and consistency distribution in the gasification processes, so it is described by 2 non-linear elliptic partial differential equations with Dirichlet boundary condition. The problem is a optimal source control problem that controls the source term necessary to approximate the temperature to a proper target function. First, we derived the optimal condition. 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 by using optimal condition of original and approximate problem. And we also evaluated the upper estimate of a posteriori error by finite element method (FEM). We proved the convergence to 0 of a posteriori error indicator (term of the right side of inequality) when division diameter converges to 0. For this, we acquired the lower bound estimation of a posteriori error and proved that a priori error and total variance error converges to 0 when division diameter converges to 0, so that we proved the convergence problem of a posteriori error indicator.

VL  - 10
IS  - 2
ER  -

Copy | Download