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

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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

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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

1. Introduction

Differential equations and their control problems are frequently found in the technical processes accompanied by energy and in the fields of optimal design. In the preliminary literatures, they derived a priori and a posteriori error estimates for the non-linear differential equations by FEM, finite volume method (FVM) and spectral method

[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.
[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.
[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). View Article
[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.

[1-4, 7-9, 14, 16-18]

, and even now for optimal control problems

[5, 6, 10-13, 15]

. When solutions of differential equations are smooth, spectral method is useful for a posteriori error estimates, otherwise, FEM is preferable. One of the difficulties in estimating a posteriori errors by FEM is to estimate the sensitivity of a posteriori error indicator to

h

, mesh size, i.e, to prove the convergence of a posteriori error indicator to zero when

h

tends to zero. That’s because, unlike by spectral method, by FEM, the error indicator

η

in estimating a posteriori error is represented by an infinite series of element-wise error indicators with respect to the total number of elements when

h

tends to zero.

{‖y - y_{h}‖}^{2} \leq η^{2} = C \sum_{k} {‖η_{k}‖}_{0, k}^{2} h_{k}^{2} = η_{Ω}^{2} h^{2},

η_{Ω}^{2} = \sum_{k} {‖η_{k}‖}_{0, k}^{2}

Therefore, upper bound estimate of a posteriori error makes sense only when error indicator converges to zero. In

[13]

, for control problems of linear differential equations, convergence to zero for error indicators was discussed under the assumption that a sequence of approximate solutions converges to analytical solution in the sense of subsequences. And in

[5]

, for semi-linear parabolic source control problems, we acquired a posteriori error estimates by spectral method but no convergence results of error indicator.

A priori and a posteriori error estimates for linear and semi-linear convection-diffusion equations were presented in

[1, 2]

. However, in case of a system of non-linear convection-diffusion equations, no error estimates and convergence results for not only equations but also their control problems might be presented at all.

The purpose of this paper is to discuss on a posteriori error estimates by FEM for source control problem governed by a system of semi-linear convection-diffusion equations and convergence of error indicator η to zero. For convergence of error indicator, we estimated lower bound of a posteriori error, a priori error and oscillation error by FEM, introduced some finite dimensional norm, and proved them by using Riesz’s theorem in Hilbert space. This paper is organized as follows.

In section 1, we presented problems to solve and optimality conditions, in section 2, a priori error estimates, in section 3 and 4, upper and lower bound estimates for a posteriori error, in section 5, convergence of a posteriori error indicator.

Here are some notations.

{‖.‖}_{s}

: norm in

{(., .)}_{s}

: inner product in

{‖.‖}_{s, K}

{‖.‖}_{s, E}

: norm in

(where K is patch of element in

, E is edge of element in

, s = 0, 1, 2).

{(., .)}_{s, K}

{(., .)}_{s, E}

: inner product in

{‖.‖}_{0, Γ} = {‖.‖}_{L^{2} (Γ)}

: norm in

L^{2} (Γ)

{(., .)}_{0, Γ}

: inner product in

L^{2} (Γ)

{‖.‖}_{0, p, K}

{‖.‖}_{0, p, E}

: norm in

L^{p} (K)

L^{p} (E)

{(., .)}_{0, p, K}

{(., .)}_{0, p, E}

: inner product in

L^{p} (K)

L^{p} (E)

Assume that

is bounded convex polygonal domain in

and

is boundary of

. The state model is as follows.

(1)

The weak solution of (1) satisfies the following equation.

(2)

[Assumption 1] Functions

satisfy the following conditions.

Under this assumption, y, g, the solutions of (2), uniquely exist in

[5]

. Cost functional is

- is known.

[Problem 1]

where y=y(f,u) is the unique solution of (2) and

[Theorem 1] (Optimality condition) Assume that

is the solution of problem 1. Then there exists a function

so that it satisfies the following system.

(3)

where

(\overset{̅}{y}, \overset{̅}{g})

stands for the unique weak solution of (2) when

f = \overset{̅}{f}

u = \overset{̅}{u}

(Proof) Let us identify

\dot{y_{f}}

\dot{g_{f}}

with Gateaux differentiation to f at

, the solution of (2).

satisfy the following equations, respectively.

(4)

(5)

Let us identify

\dot{y_{u}}

\dot{g_{u}}

with Gateaux differentiation to u at

, the solution of (2). Then

\dot{y_{u}}

\dot{g_{u}}

satisfy the following equations.

(6)

(7)

Then, using Gateaux differentiation to f, u at

(\overset{̅}{f}, \overset{̅}{u})

of cost functional, the optimality condition is

(8)

Now let us consider

{\overset{̅}{p}}_{i} \in V (i = 1, 2)

, the solution of the following conjugate system.

(9)

(10)

Let

v = p_{1}

in (4) and

v = \dot{y_{f}}

in (9), and by using assumption 1 and

, then we have

(11)

Let

v = p_{2}

in (5) and

v = \dot{g_{f}}

in (10), and by comparing 2 equalities, then we have the equality

(12)

From (11) and (12), we have

(13)

Let

v = p_{1}

in (6) and

v = \dot{y_{u}}

in (9), and by using assumption 1 and

, we have the equality

(14)

Let

v = p_{2}

in (7) and

v = \dot{g_{u}}

in (10), and by comparing 2 equalities and using

, then we have

(15)

Therefore, from (14) and (15), we have

(16)

and from (8), (13), (16), we have

It is trivial that

\overset{̅}{p_{i}} \in H^{2} (Ω)

, and we have

from the fact that embedding

H^{2} (Ω) \subset C^{0} (Ω) \subset L^{\infty} (Ω)

is continuous. Therefore, we can obtain the result of this theorem.

2. A Priori Error Estimate by FEM

Now we partition Ω into regular triangles

K_{j} (j = 1, 2, \dots, M)

, let us denote the diameter of triangle

h_{K_{j}}

, length of edge by

h_{E_{j}}

. Let

T_{h}

be the family of triangulation,

E_{h}

be the set of edges of

T_{h}

Let us denote finite element function space of

H_{0}^{1} (Ω)

where

isthelinearpolygonalspaceonelementK,

The state equation in the finite element function space is

(17)

The finite element approximation of problem 1 is as follows.

[Problem 2]:

where

is the unique solution of (17).

[Lemma 1] When

is the solution of problem 2, there exists a

so that it satisfies the following system of equations.

(Optimality system)

(18)

where

is the unique solution of (17).

(Because the proof of this lemma is similar to theorem 1, we shall omit it here.)

[Lemma 2] Assume that condition

holds true. Then the equation (18) has a unique solution, where

is the positive constant such that

(Proof) To begin with, let us consider the following iteration scheme for existence of the solution.

(19)

By using assumption 1 and

[1]

, the above iteration scheme has a unique solution.

Let us set

in every equations above.

Reflecting

, from the above iteration scheme, we have

where

Similarly, the following inequality holds

where

is constant independent of

. From the inequalities above, we have

,then

(20)

where

is constant independent of

Thus there exists an weak convergent subsequence, so that

(21)

When m approaches to infinity under (21) in (19), it can be shown that

is the solution of (18).

Now let us prove uniqueness. For 2 solutions

, let us notice

. In the first equation of (18), we fix

, subtract equations with each other and choose the trial function by

, and then we have

By applying Green’s Formula to the third term of above equality, we have

and by recalling

(assumption 1), we have

{\overset{̅}{p}}_{1 h} = 0 (Ω)

from the above equality. i.e,

Likewise, we can derive

for the second equation of (18), which completes the proof.

[Lemma 3]

[3, 4]

Let

be the orthogonal projector being

. For

, the following estimates hold

First, let us derive a priori error estimates. Let

satisfy the following system.

(22)

where

is the solution of the system of equations.

[Lemma 4]

Let

be the solutions of (3) and (18). Then the following inequality holds true.

(We denote the constants independent of

This lemma can be proved easily from optimality condition.

[Theorem 2]

Let

be the solution of (3), (18), respectively. And let us assume that

. Then the following a priori error estimate holds.

(We shall denote the constants independent of h by C in the proof.)

(Proof) First of all, let us estimate

, where

is the solution of (22). Let us notice

(23)

Second, let us estimate

This time, by letting

, and using lemma 3, assumption 1 and the above estimate, then we can write

Third, let us estimate

This time, by letting

, and using lemma 3, assumption 1, then we can write

And thus, we have

(24)

Fourth, let us estimate

Similarly to estimate

, we have

(25)

Meanwhile, subtracting the equations which

satisfy, we have

by letting a trial function by

and applying assumption 1, we have

Meanwhile, subtracting the equations which

satisfy, we have

by letting a trial function by

and applying assumption 1, we have

From the above results, we can write

(26)

where

Due to the condition of this theorem, it holds

By subtracting the equations which

satisfy, we have

by letting a trial function by

under exploiting Lipschitz continuity of

and

, we have

Similarly to estimate

, for

, we have

(27)

Meanwhile, the following inequalities hold.

(28)

From lemma 4 and (23)~(28), we have

Now let us choose

being

(29)

By summing up (23)~(29), we can complete the proof.

[Lemma 5] For the solution of (23), we have the following result.

where C is constant independent of h.

(Proof) From theorem 2, for

, the solution of (22), we have the following a priori error estimate.

(30)

Let us denote the interpolation of y by

. Because of

, we can get

(31)

and by applying triangle inequality, using (30) and (31), we have

Hence the relation

holds true. We can deduce the second result of lemma in a similar way. So we can complete the proof.

3. A Posteriori Error Estimate by FEM

Now let us assume that

satisfy the following system.

(32)

We can assume that

, the solutions of (32), exist uniquely in

[Assumption 2] Functional J is strictly convex near

and there exists a constant C such that

(33)

where

is an arbitrary point in the neighborhood of

We shall assume that assumption 1, 2 are valid in the subsequent theorems.

[Lemma 6]

Let

be the solution of (3), (17), respectively. Then we have

where

are the functions defined in (32), C>0 is the constant independent of h.

(Proof) Starting from (33), we obtain

Notice that we used optimality condition here.

From the above inequality, we have

which completes the proof.

[Lemma 7]

Let

be the solution of (18), (32), respectively. And assume that

Then the following inequality holds

where

is the jump of directional derivative on the common boundary E of two elements

where

is the jump of directional derivative on the common boundary E of two elements

where

is the jump of directional derivative on the common boundary E of two elements

where

is the jump of directional derivative on the common boundary E of 2 elements

(Proof) Step 1:

Let

First, for bilinear form

, let us prove that there is a constant

independent of v such that

Relation

holds true, so we can deduce the following inequality.

where

is the constant in the norm equality

By exploiting the assumption 1 on

, and the inequality

we can deduce the following inequality.

By adding this to

and applying lemma 3, then we can obtain the following estimation.

(34)

Step 2:

Let

. By using the assumption 1 on

, then we have

By adding this to

then we have the following estimation.

(35)

Step 3:

Let

By using the assumption on

, we can obtain

By adding this to

then we have

where

is the equivalence constant between the space

From the assumption 1, using

, we can deduce

(36)

By summing up (34)~(36), we can complete the proof.

[Theorem 3]

Let

be the solution of (3) and (18), respectively. If the condition

is valid, then we can obtain the following upper bound estimate for a posteriori error

(Proof) From lemma 6 and lemma 7, we have

(37)

Meanwhile, the following inequalities hold.

(38)

(39)

By subtracting the equation satisfying

, and choosing a trial function as

, and using assumption 1, then we have

This time, by subtracting the equation satisfying

, and choosing a trial function as

, using assumption 1 and the monotonity of

, then we have

From assumption 1 and above inequalities, we can deduce the following.

Denoting the equivalence constant of norm of

, and taking into account

and assumption 1, then it follows that

where

Similarly, subtracting the equation satisfying

, choosing a trial function as

, taking into account assumption 1, and then we have

Similarly, subtracting the equation satisfying

, choosing a trial function as

, taking into account assumption 1 and above estimations, and then we have

Due to assumption 1, all the above conditions are satisfied when

. Thus there holds

(40)

and for all v such that

, by summing up (37)~(40), we can get the desired result. This completes the proof.

4. Lower Bound Estimate for a Posteriori Error by FEM

Let us introduce the following function using barycentric coordinate

[3]

And let us introduce the following function depending on the end points (number 1, 2) of edge

, which is the common edge of element

[Lemma 8]

[3]

For every element

and edge

, b_K, b_E have the following characteristics.

[Theorem 4]

Let

be the solution of (3), (18). Then we have the following lower bound estimate for a posteriori error by FEM.

where

is independent of

and two of Cs are not the same.

(Proof) ∙ Estimation of

Let us consider the average

where

By exploiting lemma 3 and lemma 8, we have

Taking into account

, we can obtain the following estimates.

Estimation of

Let us consider the average

where

Using lemma 3 and lemma 8, we also have the following estimate.

Here

is taken into account. Likewise, considering the Lipschitz continuity of

, we can deduce the following estimates.

Similarly to the estimations of

, we can get the followings.

By summing up

, we can derive the result of this theorem. This completes the proof.

5. Convergence of a Posteriori Error Indicator

[Lemma 9] For

, oscillation error (Theorem 4),

(We shall denote the constants independent of h by the same character C in the proof.)

(Proof) Step 1: Let us consider the following inequality.

(Cisindependentofh).(41)

where

are the quantities defined in lemma 7 being

Due to coercivity of objective function J with repect to control

, it follows that

is bounded in

independently of h, i.e.

(C is constant independent of h).

Letting

in the variational equation

(42)

satisfied by

, and by using assumption 1, lemma 5 and

, then it follows

From (42), we have

Let us introduce norm in

. Because all finite dimensional norms are equivalent, it follows

Let us consider the functional

Considering the fact

is bounded linear functional in

, and from Riesz’s representation theorem, there is a unique

such that

, and it follows

Recalling assumption 1 and lemma 5,

(C is independent of h), so we have the following result.

Step 2:

(43)

(44)

(45)

where C is constant independent of h.

By summing up (43)~(45), and using the definition of osc, we can obtain the result of this theorem. This completes the proof.

Now we can state the convergence of a posteriori error indicator

[Theorem 5]

Let

be the solution of (3), (18), respectively. We have the following convergence results when

h \to 0

(Proof) We can prove 1), 2) from theorem 2, and then 3) from lemma 9, theorem 2 and theorem 4.

6. Conclusion

In this paper, for the optimal source control problem of a system of 2-dimensional semi-linear steady convection-diffusion equations, we acquired the optimality condition. And we estimated the a priori and a posteriori error and proved the convergence to 0 of a posteriori error indicator when division diameter converges to 0.

Abbreviations

FEM	Finite Element Method
FVM	Finite Volume Method

Funding

This work is supported by National Academy of Democratic People’s Republic of Korea. The authors have no relevant financial or non-financial interests to disclose.

Conflicts of Interest

The authors declare no conflicts of interest.

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.

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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

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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

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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

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@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}
}

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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  -

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ChangIl Kim

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

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JaYong Ri

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

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Song Jun Kim

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

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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

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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

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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

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@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}
}

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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  -

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