# Open Review of "A gradient-enhanced approach for stable finite element approximations of reaction-convection-diffusion problems"

Soheil Firooz, B. Daya Reddy, Paul Steinmann, Benoît Panicaud, Samuel Forest

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Soheil Firooz, B. Daya Reddy, Paul Steinmann, Benoît Panicaud, Samuel Forest. Open Review of "A gradient-enhanced approach for stable finite element approximations of reaction-convection-diffusion problems". 2025. hal-05279377

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Distributed under a Creative Commons Attribution 4.0 International License
**Identifiers**

Open Review

OAI hal-05279377

Reviewed Article

DOI 10.46298/jtcam.15788

**Licence**

CC BY 4.0

©The Authors

# Review of “A gradient-enhanced approach for stable finite element approximations of reaction-convection-diffusion problems”

©Soheil FIROOZ<fnref n="1" />, ©B. Daya REDDY<fnref n="2" />, ©Paul STEINMANN<fnref n="1" />,<fnref n="3" />, ©Benoit PANICAUD<fnref n="4" />,R, and ©Samuel FOREST<fnref n="5" />,R,E

## Review of version 1

Permalink: 2506.01873v1

**Authors** First we would like to thank the reviewers for carefully going through the manuscript and providing helpful suggestions for its improvement. Thanks to their constructive comments, we are able to present a more clear and better version of the manuscript. We have addressed all reviewer comments and revised the manuscript accordingly.

## 1 Reviewer 1 (Benoit PANICAUD)

**Reviewer** The authors propose a micromorphic-based approach for finite element stabilization of steady-state reaction-convection-diffusion equations, by gradient enhancement of an auxiliary variable. The free energy function contains an additional variable *g* penalized to remain close to the concentration *phi*, and its gradient. The well-posedness of the coupled-field approach is established, together with an error estimate. 1D and 2D numerical examples are presented showing the performance of the method compared to standard finite element formulations, for situations of varying reactivity and convection. The work has implications for chemical, biological and physical interactions involving production, motion and consumption of species.

This is an interesting paper providing an alternative stabilization technique for stiff reaction-diffusion problems. The method is original and surely deserves publication in JTCAM. I raise some discussion points implying minor revisions.

**Authors** We thank the reviewer for this comprehensive overview of the manuscript highlighting its key features.

**Reviewer** Eq. (1) is limited to steady state. Are there obstacles to the application of the method to transient problems?

**Authors** Our approach is not restricted to steady-state problems. There are no inherent obstacles to applying the method to transient problems. In fact, in (Firooz et al. 2024), of which the current work is an extension, we have considered a time-dependent formulation and presented numerical examples demonstrating the method's applicability to transient cases.

<footnote n="1">¹ Institute of Applied Mechanics, Friedrich-Alexander-Universität Erlangen-Nürnberg, Erlangen, Germany</footnote>

<footnote n="2">² Department of Mathematics and Applied Mathematics and Centre for Research in Computational and Applied Mechanics, University of Cape Town, South Africa</footnote>

<footnote n="3">³ Glasgow Computational Engineering Center, James Watt School of Engineering, University of Glasgow, United Kingdom</footnote>

<footnote n="4">⁴ Université de Technologie de Troyes, France</footnote>

<footnote n="5">⁵ MINES Paris, PSL University, MAT-Centre des Matériaux, CNRS UMR 7633, 78000 Versailles, France</footnote>

<footnote>R Reviewer</footnote>

<footnote>E Editor</footnote>
**Reviewer** Same question regarding the limitation to incompressible flow.

**Authors** Our current formulation assumes a divergence-free (incompressible) velocity field as input. This would correspond to the important case of an underlying Newtonian fluid flow governed by the Navier-Stokes equations. The assumption of compressibility would require some modifications to the analysis. In the case of a coupled problem compressible flows typically present further challenges.

**Reviewer** The most puzzling question is why the authors penalize the difference $[\nabla\varphi - g]$ instead of $[\varphi - g]$. In the present work $g$ is therefore akin to $\nabla\varphi$. As a consequence, the free energy (3) depends on $\nabla g$ which is close to $\nabla\nabla\varphi$. The theory therefore becomes a second concentration gradient theory, i.e. one order higher than the Cahn-Hilliard theory. It seems to be too much. In contrast, in (Ubachs et al. 2004), the authors penalize (phi-g), which leads to a micromorphic approach to the Cahn-Hilliard theory. The authors in (Forest and Miranville 2023; Miranville 2024) have proved the well-posedness of the corresponding boundary value problem using a closely related micromorphic approach. The higher order theory presented here seems to be the micromorphic counterpart of the grade 2 diffusion model introduced in (Temizer and Wriggers 2010). Can you tell us why the proposed theory is more advantageous than the mentioned ones akin to Cahn-Hilliard?

**Authors** The theory is based on the introduction of an auxiliary variable $g$ which is related to $\nabla\varphi$ in a weak sense in the MZAD formulation (Firooz et al. 2024), of which this work is an extension. In the current work it is possible, through the use of $g$, to formulate a gradient-enhanced theory that involves only the two variables $\varphi$ and $g$, and their gradients. In this way it becomes possible to adopt a simple finite element approach using low-order, continuous polynomials. Furthermore, we establish conditions on the choice of constants that relate the solutions of this modified formulation to the original problem. In such settings, directly introducing an auxiliary micromorphic field $g \approx \nabla\varphi$ and penalizing their difference is a natural and effective way to regularize and stabilize the discretized system, particularly under convection- or reaction-dominated regimes.

In contrast, penalizing $[\varphi - g]$ as done in (Ubachs et al. 2004; Forest and Miranville 2023; Miranville 2024) leads to a micromorphic approximation of the Cahn-Hilliard model, where the auxiliary field approximates the concentration itself. This approach is well-suited to phase separation problems governed by nonconvex free energy potentials. Our objective, however, is different; we target linear reaction-diffusion-convection problems and aim to construct a formulation that remains accurate and stable across a wide range of Péclet and Damköhler numbers. Moreover, the higher-order character of our formulation is not intended to model nonlocal physical interactions (as in phase-field models), but rather to introduce a nonlinear computational mechanism that captures steep solution gradients with minimal numerical diffusion. In that sense, the model can be interpreted as a micromorphic analogue of the grade-2 diffusion model introduced in (Temizer and Wriggers 2010), as the reviewer correctly noted. To summarize, the key advantage of our approach lies in its ability to preserve solution accuracy and suppress numerical instabilities in convection-reaction-diffusion systems by treating $\nabla\varphi$ directly as the primary quantity of interest.

**Reviewer** Is it preferable to replace the penalization $H$ of $[\nabla\varphi - g]$ by a Lagrange multiplier?

**Authors** Penalization $H$ of $[\varphi - g]$ by a Lagrange multiplier would require a quite different formulation, in which there would now be three variables to solve for, viz. $\varphi$, $g$, and the multiplier, with three governing equations. It is not clear how exactly to construct such a formulation. In the current formulation, $H$ provides an input in which, for example, the flow direction is taken into account.

**Reviewer** According to the remark after (32), increasing the penalization $H_{\text{max}}$ increases the error. Can you determine an optimal value of $H$? How do you motivate the expressions (33) for $H$ and $A$?

**Authors** Yes, using a very large value for $H_{\text{max}}$ can lead to excessive smoothing of the solution, which in turn increases the error. The expressions for the optimal values of $H$, as given in Equation (33), are motivated by the stabilization techniques proposed by Brooks and Hughes (1982) and Tezduyar and Park (1986).
**Reviewer** Looking at (33) and (34), it appears that the micromorphic approach does not leave any free parameter. Would the introduction of tunable parameters *H* and *A* be more efficient?

**Authors** In our formulation, the micromorphic parameters *H* and *A* do not have the same physical interpretation as in classical micromorphic elasticity. For example, the coupling tensor *H* cannot take arbitrarily large values, and the stiffness tensor *A* is not related to any intrinsic length scale of a microstructure. Instead, in our approach, *H* serves as a directional gradient modifier aligned with the flow field to mitigate numerical instabilities. While these parameters are not freely tunable in the conventional sense, they must be chosen carefully—following the expressions in Equation (33)—to balance stability and accuracy in the numerical solution.

**Reviewer** Can the code be provided through an appropriate github?

**Authors** As the original manuscript on the MMAD method is still under review (Firooz et al. 2025), we respectfully prefer not to share the code at this stage. We intend to make the code publicly available through an appropriate GitHub repository upon the paper's acceptance and publication.

**Reviewer** Which finite elements did you use: linear or quadratic interpolation or more?

**Authors** In all numerical examples, we employed equal-order linear elements in 1D and bi-linear elements in 2D for both fields, $\varphi$ and $g$.

**Reviewer** It seems that exact analytical solutions have been used for the 2D validation. However, I do not see the corresponding curves in Fig. 4 or 5.

**Authors** The exact analytical solution is shown only for the 1D case. For the 2D cases, instead of the exact solution to this problem, a reference solution obtained from a significantly refined mesh was used as a proxy for the exact solution. This solution exhibits very sharp gradients where the transition occurs within the width of a single element.

This reference solution is not included in the figures as the behavior is qualitatively reproduced by the MMAD method in the initial examples shown in Figures 3 and 4. Accordingly, we state that “the MMAD method leads to a highly accurate solution for the reaction-dominated case and the case with equal Damköhler and Péclet numbers,” based on the excellent agreement with the highly resolved solution.

**Reviewer** Can you compare the computation time for MMAD and FEM?

**Authors** We have added Fig. 8 and the respective explanations where we compare the computational time associated with FEM and the MMAD method in both 1D and 2D settings. Clearly, the MMAD method is more computationally expensive compared to FEM due to the additional auxiliary field we introduced.

**Reviewer** The conclusion is a bit short. Do you see some limitations of the approach?

**Authors** We have now written a more extensive set of conclusions in which we comment on extensions of the current problem to fully coupled situations in which the velocity is an additional unknown.

## 2 Reviewer 2 (Samuel FOREST)

**Reviewer** The proposed article presents theoretical approach concerning reaction-diffusion-convection problem. By using a micromorphic variable, authors bring a new solution to numerical oscillatory problem of classical Galerkin FEM. Numerical simulations are also provided for illustration. This article deserves to be published. However, there are still few questionings and improvements before to be published in JTCAM.

**Authors** We thank the reviewer for this comprehensive overview of the manuscript highlighting its key features.

**Reviewer** Page 11, line 1, “In this section we evaluate the performance the MMAD” – “In this section we evaluate the performance of the MMAD”; please change.

**Authors** The sentence has been corrected to: “In this section, we evaluate the performance of the MMAD.”
**Reviewer** Page 17, line 5, a minus sign is lacking: $u_1 = -x_2$.

**Authors** The missing minus sign has been added, and the expression now correctly reads: $u_1 = -x_2$.

**Reviewer** Page 18, line 9, the unit of diffusion coefficient is lacking. It should be in $m^2/s$.

**Authors** The unit of the diffusion coefficient has been added and is now correctly stated as $m^2/s$.

**Reviewer** Page 3, what is $F$ in equation (1)?

**Authors** As stated in the manuscript, the term $F$ in Equation (1) represents the source term in the governing equation.

**Reviewer** Page 4 line 2, concerning the reaction coefficient, which kind of reactions are concerned? If I understand well, only those for which $\varphi + X \to Y$ or $\varphi \to Y$ with specific stoichiometric factor (= 1) and total/complete reaction are concerned. For the first case, a constant concentration of $X$ should also be assumed. A precision is required to add in the manuscript.

**Authors** In our work, we consider a general form of the reaction-convection-diffusion equation, where the reaction term is linear in $\varphi$, typically written as $D\varphi$, with $D > 0$. We do not model any specific chemical reaction mechanism; instead, the reaction term is treated phenomenologically to represent linear decay or transformation processes. The formulation can be interpreted as being consistent with simplified reactions such as $\varphi + X \to Y$, or $\varphi \to Y$ under the assumption of constant concentration of species $X$, leading to an effective first-order reaction.

**Reviewer** Page 4 line 6, which kind of source term is under concern? physically, what is the difference with the reaction term? A precision is required to add in the manuscript.

**Authors** The source term $F$ represents an external input or generation term that is independent of the transported quantity $\varphi$. Physically, it accounts for any imposed production or supply of $\varphi$, such as an external injection, forcing, or emission. In contrast, the reaction term is proportional to $\varphi$ and models internal processes that consume or transform $\varphi$ over time, such as decay or conversion.

**Reviewer** Page 4, what is $t_p$ in equation (2)? A precision is required to add in the manuscript.

**Authors** The term $t_p$ denotes the boundary flux vector associated with $\nabla\varphi$, and is given by $t_p = \nabla\varphi \cdot n$, where $n$ is the outward unit normal vector to the boundary. This will be explicitly clarified in the revised manuscript.

**Reviewer** Page 5, from a mathematical point of view, what is the consequence for the choice of a non-quadratic expression such as $F\varphi$ in equation (3)? A precision is required to add in the manuscript.

**Authors** The term $F\varphi$ corresponds to the potential energy associated with the source $F$, which is conservative and does not depend on $\varphi$. The linear dependence on $\varphi$ is standard, as opposed say to the energy associated with diffusion which is quadratic in $\nabla\varphi$. Minimization of the energy in (3) leads to the weak form (6). If we were to apply a reverse integration by parts to $(6)_1$ this would give the governing convection-diffusion equation with right-hand side (source term) equal to $F$.

**Reviewer** Page 5, you subtract $F\varphi$ in equation (5). According to the definition of $\Psi_0$, the resulting term should not then depend on $F\varphi$. So how this term could appear in equation (6)? A precision or correction is required to add in the manuscript.

**Authors** The term $F\varphi$ in Equation (5) is indeed redundant and should be omitted, as it is already accounted for in the definition of $\Psi_0$. We will correct this in the revised manuscript accordingly.

**Reviewer** Page 5, you mention just before equation (6) that “is then added the convective term”. What is the justification to add it? Is there possibility to deduce it from a weak formulation? From my comprehension of this work, the restriction is due to the fact/choice that velocity $U$ is not an output/unknown to solve of the problem, but an input. A precision is required to add in the manuscript.

**Authors** In our formulation, we adopt an energy-based approach to derive a variational form of the reaction-convection-diffusion equation. For the diffusion and reaction terms, appropriate energy
contributions can be directly constructed. However, the convective term poses a challenge since it is inherently non-self-adjoint and does not directly arise from an energy functional. Thus, we construct the energy functional corresponding to the reaction-diffusion problem and derive the weak form from this; then we add to the weak form the convective term.

**Reviewer** Page 7, third equation in equation (11), for continuity of $\ell(\cdot)$, would not it be symbol $\le$ instead of $=$?

**Authors** Indeed, it should be $\le$. Thank you for pointing this out.

**Reviewer** Page 7, please check equation (13). I do not clearly understand the right member. Factors 2 may be missing somewhere.

**Authors** While Equation (13) is mathematically correct as originally stated, we agree that its structure may be unclear at first glance. To improve clarity, we have reformulated the expression in a more transparent form in the revised manuscript, ensuring that all coefficients and terms are explicitly visible.

**Reviewer** Page 8, please check the second equation of equation (18). I do not understand neither the missing of Da term, nor the lack on integration symbol for second part for $|g_h|^2$ terms, because it is only in last equation that integration is included in the norm definition $||g_h||_G^2$.

**Authors** Regarding the omission of the Damköhler number (Da) in the second equation of Equation (18) (Eq. (17) in the current version), this is intentional. The inequality is used to estimate an upper bound $B$, and since the omitted term with the Damköhler number is non-negative, removing it from the right-hand side does not violate the inequality. In the second line the terms referred to should read $||g_h||_{L^2}$ and $||\nabla g_h||_{L^2}$. This was an error and the referee is thanked for pointing this out.

**Reviewer** Page 8, please check the first equation of equation (21). Where is the factor Da? Also please add somewhere in the manuscript the definitions of norm $||g_h||_{L^2}$ and $||\nabla g_h||_{L^2}$ occurring in the second equation of equation (21).

**Authors** Indeed, in the first equation of Equation (21) (Eq. (20) in the revised version), the factor Da is missing from the immediate term following the inequality. This has been corrected in the revised manuscript. The definitions of the norms in Eq. (7) are now expanded to include $||g_h||_{L^2}$ and hence $||\nabla g_h||_{L^2}$.

**Reviewer** Page 9, line 1, for consistency and completeness of the article, I suggest to give the proof for continuity of $\ell(\cdot)$. Moreover, the expression of $c$ in the third equation of equation (11) has to be explicit. Maybe the proof for continuity of $\ell(\cdot)$ can be added in an appendix.

**Authors** The proof of continuity of $\ell(\cdot)$ is fairly short and is now included.

**Reviewer** Page 9, line 5, is the constant $c$, the same as in the third equation of equation (11)? Whatever the answer, a precision is required to add in the manuscript.

**Authors** The constant $c$ mentioned on page 9, line 5, is not the same as the constant $C$ appearing in the third equation of Equation (11). To avoid confusion, we clarify in the manuscript that the relationship between $C$ and $c$ is provided immediately after Equation (24).

**Reviewer** Page 9, maybe you could precise somewhere that equation (25) corresponds to $H = 0$. In other words, the original problem for $\varphi_0$ corresponds to $H = 0$, that is why your bilinear form is now denoted $a$ and no more $B$. A precision is required to add in the manuscript. Is $a(\cdot, \cdot) + b(\cdot, \cdot) = B(\cdot, \cdot)$ in equation (26)?

**Authors** As explained in the manuscript, Equation (25) corresponds to the original formulation of the problem in which no micromorphic auxiliary variable is introduced. The tensor $H$ does not appear at all in this formulation because the micromorphic regularization is not yet applied. It is not the case that $a(\cdot, \cdot) + b(\cdot, \cdot) = B(\cdot, \cdot)$. In fact (9) corresponds to the pair of equations in (8), so that $B(\cdot, \cdot)$ is the left-hand side of this pair. On the other hand, (26) corresponds to the first of equations (8), the left-hand side of which is contained in the terms $a(\cdot, \cdot) + b(\cdot, \cdot)$.
**Reviewer** Page 10, line 5 in the remark, you should explicit the origin of your equality $\beta_0/M_0 = H_{\max}Pe$. Could you also develop and clarify/justify the sentence "The finite element approximation error depends on $m/M$, where these constants are given in (23) and (19).", please? The dependence that you propose is not obvious.

**Authors** The relation $\beta_0/M_0 = H_{\max}Pe$ is derived by combining the expressions that immediately follow Equations (28) and (30), where $M_0$ is shown to be proportional to the inverse of the Péclet number and $\beta_0 = H_{\max}$ is proportional to $H_{\max}$. Regarding the second point, the sentence "The finite element approximation error depends on $m/M$" refers to the general theory of finite element stability and convergence, where $M$ denotes the coercivity constant and $m$ the continuity constant of the bilinear form (as defined in Equations (22) and (18), respectively). In standard a priori error estimates, the ratio $m/M$ influences the constants appearing in the error bounds. This is indicated in the line following (24), with a reference to Reddy 1998 where the details maybe found.

**Reviewer** Page 10, I suggest to remind explicitly in the text the meaning of $B$, $D$ and $h$. What is the difference between $h$ and $h_i$? Is $h^2$ the norm of $h_i$? A precision is required to add in the manuscript.

**Authors** We will revise the manuscript to explicitly recall the definitions of $B$, $D$ and $h$ where they appear. Specifically, $h_i$ refers to the characteristic element size in the i-th coordinate direction. Since we employ structured meshes in all numerical examples, the mesh is uniform in all directions, and thus $h_i = h$ for all $i$. In the case of unstructured meshes, defining an appropriate scalar representative mesh size becomes more involved. There is a body of literature addressing optimal definitions or averaging strategies for $h$ in such cases. We will briefly note this distinction in the manuscript and clarify that the current expressions assume structured meshes, as used in our computations.

**Reviewer** Page 11, line 9, why do you choose $F = 1$ for example 1? Physically, to what does it correspond? A precision is required to add in the manuscript.

**Authors** In Example 1, we set $F = 1$ as a constant source term of unit magnitude. This choice is made purely for simplicity and normalization purposes. It allows us to clearly observe and distinguish the behavior of the solution under varying Damköhler and Péclet numbers, without introducing additional complexity from a spatially varying or parameter-dependent source.

**Reviewer** Page 11 in figure 1, example 4, there is twice $\varphi = 0$ for bottom condition. Is this relevant?

**Authors** The repeated boundary condition $\varphi = 0$ on the bottom boundary in Example 4 of Figure 1 is indeed an error. We will correct the figure by removing the redundant condition in the revised manuscript.

**Reviewer** Page 13, line 2, can you give more details on the meshing. Is it a regular/structurated one or not? Please also provide information for the 2D domain.

**Authors** We thank the reviewer for the request for clarification. In both the 1D and 2D cases, structured (regular) meshes are used. For the 1D examples, the domain is discretized into 100 linear elements. For the 2D examples, the domain is discretized into a structured grid consisting of 1,600 bi-linear quadrilateral elements. In the MMAD method, equal-order linear (1D) and bilinear (2D) approximations are employed for both $\varphi$ and $g$. We will add this information to the revised manuscript for clarity.

**Reviewer** In general, in this simulation part, can you also give more information about numerical features, such as calculation time?

**Authors** The implementation is carried out in MATLAB using a standard finite element framework. We have added Fig. 8 and the respective explanations where we compare the computational time associated with FEM and the MMAD method in both 1D and 2D settings. Clearly, the MMAD method is more computationally expensive compared to FEM due to the additional additional auxiliary field we introduced.

**Reviewer** Page 14, line 5, why do you choose $F = 0$ for example 2? Physically, to what does it correspond? A precision is required to add in the manuscript.
**Authors** In Example 2, the choice $F = 0$ is made to replicate the conditions studied in (Brooks and Hughes 1982), allowing for a direct comparison with their results. Physically, this corresponds to a case with no external source term, where the behavior of the system is governed purely by convection, diffusion, and reaction processes.

**Reviewer** Page 14, line 5, you mention that “the flow is irrotational with angle 45°”. It is an imposed condition through the values of dimensionless velocity **u**? A precision is required to add in the manuscript.

**Authors** Yes, the statement that “the flow is irrotational with angle 45° refers to how the velocity field is defined, as it is an input to the problem. Specifically, the components of the dimensionless velocity **u** are chosen such that the flow direction forms a 45° angle with the coordinate axes.

**Reviewer** Page 18, line 6, why do you choose $F = 0$ for example 6? Physically, to what does it correspond? A precision is required to add in the manuscript.

**Authors** In Example 6, the choice $F = 0$ is made in order to reproduce the setting considered in (Codina 1998), allowing for meaningful comparison with their results. Physically, this corresponds to a scenario with no external source term, where the dynamics are governed solely by convection, diffusion, and reaction effects.

## References

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Codina, R. (1998). Comparison of some finite element methods for solving the diffusion-convection-reaction equation. *Computer Methods in Applied Mechanics and Engineering*. DOI: 10.1016/S0045-7825(97)00206-5.

Firooz, S., B. D. Reddy, and P. Steinmann (2025). A micromorphic-based artificial diffusion method for stabilized finite element approximation of convection-diffusion problems. DOI: 10.48550/arXiv.2506.14800.

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Tezduyar, T. E. and Y. J. Park (1986). Discontinuity-capturing finite element formulations for nonlinear convection-diffusion-reaction equations. *Computer Methods in Applied Mechanics and Engineering*. DOI: 10.1016/0045-7825(86)90003-4.

Ubachs, R. L. J. M., P. J. G. Schreurs, and M. G. D. Geers (2004). A nonlocal diffuse interface model for microstructure evolution of tin-lead solder. *Journal of the Mechanics and Physics of Solids*. DOI: 10.1016/j.jmps.2004.02.002.

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