A Posteriori Error Estimates and Convergence of Error Indicator by FEM for a Semi-linear Elliptic Source-boundary Control Problem

Chang Il Kim; Jong Hyok Kang; Gi Chol Sok

doi:doi:10.11648/j.ml.20251102.12

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A Posteriori Error Estimates and Convergence of Error Indicator by FEM for a Semi-linear Elliptic Source-boundary Control Problem

Chang Il Kim^*, Jong Hyok Kang, Gi Chol Sok

Published in Mathematics Letters (Volume 11, Issue 2)

Received: 24 January 2025 Accepted: 17 July 2025 Published: 3 September 2025

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Abstract

In this paper, we obtain convergence of a posteriori error indicator to 0 when the mesh size h goes to 0 for the finite element approximation of source-boundary control problems governed by a system of semi-linear elliptic equations. We give the upper and lower bound of a posteriori error, and convergency of a posteriori error indicator.

Published in	Mathematics Letters (Volume 11, Issue 2)
DOI	10.11648/j.ml.20251102.12
Page(s)	41-59
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

Semi-linear Elliptic Equations, Source Control, A Posteriori Error Estimate

1. Introduction

Differential equations and their control problems appear a lot in the thermal engineering, metallurgical engineering, chemical process and in the field of a system of optiamality. In many previous works, a posteriori error estimates of a differential equation have been done by using finite element method and spectral method

[1-3, 7-9, 14, 16-18]

, and these methods are being extended to optimal control problems

[4-6, 10-12, 13, 15]

If the solution of a differential equation is smooth, a posterior error estimates are obtained by spectral method

[4, 5]

but if not, by finite element method

[6, 10, 12, 13, 15]

. One of the difficulties in estimating a posteriori error by finite element method(FEM) is to verify the convergence of a posteriori error indicator to 0 when the mesh size h goes to 0, for an error indicator

is denoted as the form of sum with respect to elements of an indicator

, i.e., as the form of infinite series when h goes to 0 unlike spectral method.

[12]

, for control problem of a linear differential equation, the convergence of an error indicator was studied in the meaning of a subsequence. In

[5]

, for a semi-linear parabolic source control problem, the upper bound estimates of a posteriori error by spectral method were obtained but not the convergence of an indicator.

The purpose of this work is to obtain a finite element posteriori error estimates for a semi-linear elliptic source control problem and study the convergence of an error indicator.

We are not aware of any existing work on a posteriori error estimates and the convergenc of error indicator by FEM for a semi-linear elliptic control problem.

For convergence of an error indicator, we’ve obtained a finite element posterio-ri lower estimates, estimated a priori error and osc (osc: oscilliation) of the residual, introduced a proper finite dementional norm and applied proof method using Lis theorem.in the Hilvert space.

Our paper is constructed as follows. In section 2, we give establishiment of problem and optimality condition, in section 3, 4 upper, lower bound of a posteriori error estimates of finite element, in section 5, priori error estimates and in section 6, convergency of error indicator of FEM.

Let

denote the

norm and inner product. We set

with

.We denote by

the

norm and inner product, and by

the

-norm and duality product between

and

2. Setting of Problems and Optimality Condition

Let

be a bounded convex polygonal domain in

with boundary

Let us consider the boundary problem of differential equation as follows.

(1)

are to be controlled and the feasible sets are convex closed subsets of

as follows.

[Assumption 1] We assume the following properties for

and they are strictly monotonic in

The weak form of model (1) is (2) as follow

(2)

We assume that there exist unique weak solutions

of (2) in

The cost functional is defined as follows:

where

is given,

[Problem 1]

where

is a solution of (2).

Assume that there exists at least one solution of Problem 1.

[Theorem 1] (Optimality conditions)

Assume that

is a solution of Problem 1. There exists

that satisfies:

(3)

where

are unique solutions of (2) when

Proof. Let

denote Gateaux differential of the solution of (1)

in the direction of

in the

in the direction of

satisfy the following equations, respectively.

(4)

The Gateaux differentials of the cost functional at

in the direction of

are as follows:

Let us consider the unique solution

of the following adjoint system.

By setting

in the first equation of (4) and

in the adjoint system, and comparing them, we get

Similarly, by setting

in the second equation of (4) and

in the adjoint system, and comparing them, we get

Thus, we have

The proofis completed.

3. Upper Bound of a Posteriori Error

Now, we partition

into regular triangles

and denote the diameter of a triangle

, respectively. We set

, let

be the family of triangles,

the set of all the sides of the triangles and denote finite element function space of

as:

where

denotes a polynomial space whose order is less or equal than one on element

(5)

The finite element approximation of (2) is as follows:

Problem 2.

where

is the unique solution of(5).

Lemma 1. Problem 2 has the unique solution

, which satisfies the following system (optimality system)

(6)

(7)

where

is the unique solution of (7).

Proof. It’s similar to the proof of Theorem 1.

Lemma 2 (

[2, 3]

) Let

denote an orthogonal projection operator. For

, we have:

Lemma 3

1) If

is the solution of problem 1, then

2) The solutions

of (6), (7) are bounded independent of

(proof) 1) From theorem 1, we have

and this is equivalent to the followings.

But

and

Since

by assumption 1,

from the smoothness of the solution of an elliptic equation and

has a unique solution in.

From

, the continuity of

and

and the properties of operator

, the assertion of the theorem holds.

2) From the fact that the functional

is coercive with respect to

are bounded independent of

. In the below equation satisfied by

when considering the asumption 1, we get

when

Here,

are norm equivalent constants of spaces

and

is a constant independent of

In the adjoint equation

considering assumption 1, we get

when

Let us estimate the upper bound of a posteriori error.

The following expression holds. (

[5]

)

Here,

satisfies the following system

(8)

Where

is a unique solution of the first expression of (5).

[Assumption 2] Functional is strictly convex in the neighborhood of a solution and there exists a proper constant so that

(9)

is an arbitrary element in the neighborhood of

In the future, assumptions 1, 2 are assumed to be satisfied.

Lemma 4 When are the solutions of (3) and (6), (7) respectively, it holds that

where

is a function defined in (8),

is a constant independent of

and

(proof) From assumption 2 (expression (9)), we have

Here, we consider that

from the optimality condition.

From the above inequality,

Thus, we come to the following conclusion

Lemma 5 Let be solutions of (6), (7) and (8), respectively. Then it holds that

where

are as follows.

And

is the jump value of the directional derivative in the common boundary

of two elements

Similarly,

is the jump value of the directional derivative in the common boundary

of two elements

(proof) Step 1: We set,. Considering the assumption 1 about, we get

Here,

are positive, seen in theorem 1.

Let us make a use of lemma 2 adding

to the above expression.

Considering

and 1) of lemma 2, we get

Thus, we get the following inequality.

Step 2: We set

Taking into account the assumption 1, we get the following estimation.

where

is a strictly monotonic constant of in assumption 1.

Adding

to the above expressing and using lemma 2, then we get

Therefore, we get

and the following holds.

In the following theorem, we give the upper bound estimate of a posteriori errorwhich is one of the main results of this paper.

[Theorem 2] If

are the solutions of (3) and (6), (7) respectively, then the following relation holds.

(proof) From lemma 4 and 5, it holds that

(10)

On the other hand, it holds that

(11)

(12)

Subtracting the equation satisfied by

, setting a test function as

and using the strong monotonicity of

and the monotonicity of

, we get

From above, we get the following inequality

.(13)

Similarly, subtracting the equation satisfied by

, setting a test function as

and considering assumption 1, we get

(14)

From (10) - (14), we get the result.

4. Lower Bound of a Posteriori Error

We introduce the following function by barycentric coordinates

of element

[2]

We introduce the following function when

is a common edge of elements

Lemma 6 (

[2]

) For each element

and edge

have the following properties.

[Theorem 3] For the lower bound of a finite element posteriori error, the following estimation holds.

where is independent of and denotes the different constants in the proof below, and is oscilliation

(proof) Let us estimate

It holds that

Using 2) and 3) of lemma 2, we estimate the above integrals.

From the above relations, we get the following result.

Let us estimate

1) In case of

Using inverse inequality, we estimate the above integrals

Similarly, we get the following expression.

Taking into account the above expressions, we get

2) In case of

Like in 1), we get

We estimate the above integral parts.

Here, we consider that

Thus, we get the following result.

Let us estimate

Thus, we get

Let us estimate

1) In case of

For convenience, we set

Considering the calculating process of, we can see that the following expressions hold.

2) In case of

In this case, we set

Let us estimate the above integral parts.

We estimate

. For convenience, we accept the following notation

It is obvious that the inequality

holds. We give a similar consideration like before.

Here, we used the inequality

We estimate

By estimating the above integrals, we get

By summing

, we get the result of the theorem.

5. A Priori Error Estimates

For convenience, we set the cost functional as follows

Then we get the following. (

[5]

)

where

is the solution of the following equation.

Lemma 7 Let

be the solutions of (3) and (6), (7) respectively. Then it holds that

where

is a constant independent of

(proof) From (3) and (6), (7) the relation which satisfy is as follows.

(15)

(16)

We choose

and fix

Let

be a standard interpolation operator.

By the smoothness of a control (lemma 3),

and we define

(17)

Then we can see that

In (15) and (16), we set

add together and put in order, then we get

(18)

From (18), we get inequality

(19)

Setting

and considering the boundedness (lemma 2) of finite element solutions independent of

, we get the estimation.

(20)

From (18), (19) and (20), we get the result.

The following theorem about a priori error estimates shows the convergence of an approximation solution.

[Theorem 4] Let be the solutions of (3) and (6), (7) respectively. Then it holds that

We denote by

the different constants independent of

(proof)

We set

We first estimate

. Taking into account the assumption 1, we get

where

is positive seen in assumption 1.

Considering the Lipschitz continuity of

and 1) of lemma 2, we get

(21)

We estimate

This time we set

Considering the assumption 1, strong monotonicity of, Lipschitz continuity and 1) of lemma 2, we get

(22)

On the other hand, subtracting the equations which

satisfy, it holds that

and when we set

and consider the strong monotonicity of

and monotonicity of

, then we get

(23)

Similarly, subtracting the equations which

satisfy, we get

Now when we consider

and Lipschitz continuity of

and set a test function as

, we get the following expression

where

is positive seen in assumption 1.

(24)

On the other hand, it holds that

(25).

From lemma 7 and (21)-(25), using the above results, it holds that

We choose

to satisfy

. Then it holds that

(26)

and considering (21)-(26), we get the result of the theorem. Here,

is a constant independent of

6. Convergence of a Posteriori Error Indicator

We see the following lemma before the convergence of a posteriori error indicator

Lemma 8 For

, it holds that

(In the proof, we denote by

all the constants independent of

(proof) Step 1: Let us see

(

is independent of

). Here,

are defined in lemma 5 to be

From lemma 3 and theorem 4, we get the following result about the boundedness, i.e.,

(27)

where

is a constant independent of

Thus,

and by assumption 1,

From the variational equation satisfied by

we get

We introduce the norm in

Since finite dimensional norms are equivalent, we get.

, we think of a functional

Since

is a bounded linear functional in

and by Riesz’ representation theorem, there exists a unique element

such that

And it holds that

(28)

Since

(

is independent of

), we get

Since

it holds that

Considering (28), we get

in a similar way and in turn,

Step 2: Let us see

(

is a constant independent of

). Here,

are defined in lemma 5 to be as follows.

From (27), it holds that

From an adjoint equation

it holds that

, (

are independent of

)

, introducing a functional

Then it holds that

And, similar to step 1, there is a unique element

such that

and

. Therefore, it holds that

and the following result follows.

Step 3: Let us see

Using lemma 3.4 in

[9]

(or theorem 3.1 in

[8]

), it holds that

(29)

By lemma 1 and 3, we get

and since

, we get.

where a partial derivative is a weak derivative, i.e.,

Considering the above and (27), we get

Considering (27) and (29), we get

In a similar way, we get

and thus we get the following result.

Now we give the theorem on the convergence of a posteriori error indicator which is one of the main results in the paper.

[Theorem 5] Let

be the solutions of (5) and (6), (7) respectively. When

, the following convergence result about a posteriori error indicator holds.

(proof) From lemma 3, all the constants in a posteriori error and a priori error estimates are bounded independent of

. Using theorem 4 and lemma 8, the result follows from theorem 3.

7. Conclusion

We obtained the optimality condition about a mixed control problem (two variable control) of a semi-linear elliptic equation with a control in the right side of the boundary condition and in the source term, and based on this, got a posteriori error estimates of finite element approximation solution in an adaptive finite element function space and verified the convergence of an error indicator to 0 when the meshsize goes to 0.

Abbreviations

FEM	Finite Element Method

Funding

The authors have no relevant financial or non-financial interests to disclose.

Conflicts of Interest

The authors declare no conflicts of interest.

References

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Kim, C. I., Kang, J. H., Sok, G. C. (2025). A Posteriori Error Estimates and Convergence of Error Indicator by FEM for a Semi-linear Elliptic Source-boundary Control Problem. Mathematics Letters, 11(2), 41-59. https://doi.org/10.11648/j.ml.20251102.12

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ACS Style

Kim, C. I.; Kang, J. H.; Sok, G. C. A Posteriori Error Estimates and Convergence of Error Indicator by FEM for a Semi-linear Elliptic Source-boundary Control Problem. Math. Lett. 2025, 11(2), 41-59. doi: 10.11648/j.ml.20251102.12

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AMA Style

Kim CI, Kang JH, Sok GC. A Posteriori Error Estimates and Convergence of Error Indicator by FEM for a Semi-linear Elliptic Source-boundary Control Problem. Math Lett. 2025;11(2):41-59. doi: 10.11648/j.ml.20251102.12

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@article{10.11648/j.ml.20251102.12,
  author = {Chang Il Kim and Jong Hyok Kang and Gi Chol Sok},
  title = {A Posteriori Error Estimates and Convergence of Error Indicator by FEM for a Semi-linear Elliptic Source-boundary Control Problem
},
  journal = {Mathematics Letters},
  volume = {11},
  number = {2},
  pages = {41-59},
  doi = {10.11648/j.ml.20251102.12},
  url = {https://doi.org/10.11648/j.ml.20251102.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ml.20251102.12},
  abstract = {In this paper, we obtain convergence of a posteriori error indicator to 0 when the mesh size h goes to 0 for the finite element approximation of source-boundary control problems governed by a system of semi-linear elliptic equations. We give the upper and lower bound of a posteriori error, and convergency of a posteriori error indicator.
},
 year = {2025}
}

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TY  - JOUR
T1  - A Posteriori Error Estimates and Convergence of Error Indicator by FEM for a Semi-linear Elliptic Source-boundary Control Problem

AU  - Chang Il Kim
AU  - Jong Hyok Kang
AU  - Gi Chol Sok
Y1  - 2025/09/03
PY  - 2025
N1  - https://doi.org/10.11648/j.ml.20251102.12
DO  - 10.11648/j.ml.20251102.12
T2  - Mathematics Letters
JF  - Mathematics Letters
JO  - Mathematics Letters
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PB  - Science Publishing Group
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AB  - In this paper, we obtain convergence of a posteriori error indicator to 0 when the mesh size h goes to 0 for the finite element approximation of source-boundary control problems governed by a system of semi-linear elliptic equations. We give the upper and lower bound of a posteriori error, and convergency of a posteriori error indicator.

VL  - 11
IS  - 2
ER  -

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Author Information

Chang Il Kim

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

Contact Email
Jong Hyok Kang

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

Contact Email
Gi Chol Sok

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

Contact Email

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APA Style

Kim, C. I., Kang, J. H., Sok, G. C. (2025). A Posteriori Error Estimates and Convergence of Error Indicator by FEM for a Semi-linear Elliptic Source-boundary Control Problem. Mathematics Letters, 11(2), 41-59. https://doi.org/10.11648/j.ml.20251102.12

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ACS Style

Kim, C. I.; Kang, J. H.; Sok, G. C. A Posteriori Error Estimates and Convergence of Error Indicator by FEM for a Semi-linear Elliptic Source-boundary Control Problem. Math. Lett. 2025, 11(2), 41-59. doi: 10.11648/j.ml.20251102.12

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AMA Style

Kim CI, Kang JH, Sok GC. A Posteriori Error Estimates and Convergence of Error Indicator by FEM for a Semi-linear Elliptic Source-boundary Control Problem. Math Lett. 2025;11(2):41-59. doi: 10.11648/j.ml.20251102.12

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@article{10.11648/j.ml.20251102.12,
  author = {Chang Il Kim and Jong Hyok Kang and Gi Chol Sok},
  title = {A Posteriori Error Estimates and Convergence of Error Indicator by FEM for a Semi-linear Elliptic Source-boundary Control Problem
},
  journal = {Mathematics Letters},
  volume = {11},
  number = {2},
  pages = {41-59},
  doi = {10.11648/j.ml.20251102.12},
  url = {https://doi.org/10.11648/j.ml.20251102.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ml.20251102.12},
  abstract = {In this paper, we obtain convergence of a posteriori error indicator to 0 when the mesh size h goes to 0 for the finite element approximation of source-boundary control problems governed by a system of semi-linear elliptic equations. We give the upper and lower bound of a posteriori error, and convergency of a posteriori error indicator.
},
 year = {2025}
}

Copy | Download

TY  - JOUR
T1  - A Posteriori Error Estimates and Convergence of Error Indicator by FEM for a Semi-linear Elliptic Source-boundary Control Problem

AU  - Chang Il Kim
AU  - Jong Hyok Kang
AU  - Gi Chol Sok
Y1  - 2025/09/03
PY  - 2025
N1  - https://doi.org/10.11648/j.ml.20251102.12
DO  - 10.11648/j.ml.20251102.12
T2  - Mathematics Letters
JF  - Mathematics Letters
JO  - Mathematics Letters
SP  - 41
EP  - 59
PB  - Science Publishing Group
SN  - 2575-5056
UR  - https://doi.org/10.11648/j.ml.20251102.12
AB  - In this paper, we obtain convergence of a posteriori error indicator to 0 when the mesh size h goes to 0 for the finite element approximation of source-boundary control problems governed by a system of semi-linear elliptic equations. We give the upper and lower bound of a posteriori error, and convergency of a posteriori error indicator.

VL  - 11
IS  - 2
ER  -

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