Review Reports

Round 1

Reviewer 1 Report

1. Follow the journal style.

2. Extend the introductory section.

3. I see too much abbreviations, it would be better to add a list of abbreviations.

4. Explain why the inequalities in Page 2 are important in the sequel work.

5. Cite a Ref for 2.7, Gabreial Lord textbook is welcome.

6. Support the presentation with 2D figures.

7. Reduce the number of Refs.

8. Proofread the text.

Minor editing of English language required.

Author Response

Please see the attached file.

Author Response File: Author Response.pdf

Reviewer 2 Report

·       The bibliographic references cover the topic of the article but it is necessary to add some articles from the journal . 

 

·       The paper is well written from a mathematical point of view .

I asked the authors to specify a practical application of the presented method 

Author Response

Please see the attached file.

Author Response File: Author Response.pdf

Reviewer 3 Report

1) Page 2, line 9. 'regularity requirements of the solution'.

I guess, I do not understand this sentence properly. Is it better to say 'regularity of the parameters' or 'regularity of the equation'? Or you mean that only regularity of the solution is important, no matter what are the coefficients?

2) Page 2, line -13. This question is related to we previous one. Do we require anything about f and u_0? I reckon, they must be at least functions of the correpsponding Sobolev spaces.

3) MAJOR REMARK I do not understand Condition (1.4). If we take k=0, we obtain that u_0=0. Of course, I understand that this situation can be obtained from the initial equation by a transformation of variables.

4) The line next to Eq. (1.1). I think, it is necessary to clarify what is mesh parameters.

5) Page 2, line -5. Evidently, \alpha in the sum is not the same \alpha as above. I request authors to use another letter.

6) Page 2, line-3. What is C?

7) Page 3, line 6. The function v is of a Sobolev space, so its divergence is a distribution. Do we assume that this distribution is in fact a function of L^2?

8) Next line. Probably, there is a misprint in the integral. Do we integrate over the Lebesgue measure?

9) Equations (2.1) - (2.3). We find a pair of p and u. Does it mean that we find u and define p as above? What about the uniqueness of the solution? Or we take any solution of the problem? Similar questions are applicable to Eq. (2.4)-(2.6), (2.9)-(2.9) etc.

10) Page 5, last line of Section 2. '... the integration ... is reduced greately...'.
This is true but needs to be clarified.

11) Lemma 3.1. Please clarify what is L1 scheme.

12) Lemma 3.2 'satisfies' --> 'satisfy'.

13) Eq. (3.11) The projection operator P_h has already been defined above. Is this the same operator? 
   

Author Response

Please see the attached file.

Author Response File: Author Response.pdf

Round 2

Reviewer 3 Report

I am satisfied with the corrections made.