Efficient Discretization of the Laplacian: Application to Moving Boundary Problems

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

Please, find some comments bellow.

Comments for author File: Comments.pdf

Author Response

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Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

• What is the main question addressed by the research? An efficient approximation is developed for the Laplacian operator by merging the advances of finite difference and finite element approximations. The method works on a general quadrilateral grid.
• Do you consider the topic original or relevant to the field? Does it
address a specific gap in the field? The true power of the method becomes clear in its application to moving boundary problems.
• What does it add to the subject area compared with other published
material? The proposed algorithm is a vectorized procedure keeping the computational time at a low level.
• What specific improvements gave authors consider regarding the
methodology? A moving grid is introduced which is necessary for an efficient
simulation. In the framework of the proposed algorithm, the spatial discretization on the new grid is quickly computed. The procedure is tested in the Stefan problem.
• Are the conclusions consistent with the evidence and arguments presented
and do they address the main question posed? The conclusions are consistent and address the main question posed.
• Are the references appropriate? The reference list is appropriate. The template of references has to be checked.

Author Response

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Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

While the paper discusses an optimization-based FD discretization, the applicability could be better justified. Adding context on how this method represents practical applications or specific physical properties would clarify the method choices and improve the study's relevance.

Please highlight why this method is worth using compared to other numerical solution methods.

Compare with solutions obtained by other methods by means of solutions to concrete examples of problems.

Analyse the advantages and disadvantages of the method.

Is the method applicable when the nonlinearity in the differential equation is in the derivatives?

Comments on the Quality of English Language

The quality of the English language is almost OK. It cen be polished at some places.

Author Response

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Author Response File: Author Response.pdf

Reviewer 4 Report

Comments and Suggestions for Authors

An spatial discretization method of the Laplace operator in general quadrilateral mesh was proposed in this study.  Although the paper was well organized, I cannot find the significance of this method. 

comments/questions:

1. compared to solving the problems on a transformed/mapped computational domain (uniform), what's the advantage of this proposed method. 

2. compared to the schemes obtained from direct discritization on irregular quadrilateral mesh, what is the advantage of this proposed method. 

3. second order schemes were used in this study, how about the high order schemes?

Author Response

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Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

The authors made significant corrections according to the reviewer recommendations.

In this form, the paper can be published.