A Comparison of Parallel Algorithms for Numerical Solution of Parabolic Problems with Fractional Power Elliptic Operators

Round 1

Reviewer 1 Report

Fractional differential equations with fractional powers of elliptic operators have attracted much attention during last decade. Nowadays, the development of numerical schemes for such equations is a very popular but important problem. Due to nonlocality of fractional differential operators, the numerical algorithms for fractional order differential equations usually have much more computational complexity than for integer order ones. The development of efficient parallel algorithms for such equations is a challenging problem. The manuscript under review is devoted exactly to this problem. In general, the manuscript is well written and will be interesting for readers. It is appropriate for publication in Axioms after a minor revision in accordance with the comments given below.

1. It is obvious that in the limiting case of alpha=1 the scheme (6) has the second order of approximation. But it is not clear for me why this scheme has the same order of approximation for all values of alpha. The proof of this statement or an appropriate reference is needed.

2. In general case, the fractional Laplacian is defined as a hypersingular integral. As a result, there may be a singularity in the solution of the diffusion equation with such operator even when the Dirichlet boundary condition is stated. Can such solution be found by the proposed algorithms? A brief discussion of this situation is needed. Also, the appropriate class of functions for the source function F in (5) should be explicitly defined. Say, can I take a trigonometric approximation of the Dirac delta-function as F?

Recommendation: Minor revision.

 

 

 

Author Response

Thank You for your remarks and comments. 

Author Response File: Author Response.pdf

Reviewer 2 Report

 Review report\\
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  Manuscript ID:   axioms-1596120. A comparison of parallel algorithms for numerical solution of
parabolic problems with fractional power elliptic operators.
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\vspace{0.5 cm}  The paper is devoted to  analyzing the efficiency and accuracy of general parallel solvers for three dimensional parabolic problems with a fractional power of the elliptic operator. The paper is well-written in accordance with the classical scientific style. However, the major remark is in the following. There is no sufficient comparison with the previously obtained analytical methods, for instance the methods, how to find a solution of the problem (5) as well as (32), can be found in the papers:    Kukushkin M.V.  Natural lacunae method and Schatten-von Neumann classes of the convergence exponent. \textit{arXiv},  arXiv:2112.10396v2 [math.FA],
   Kukushkin M.V.     Evolution Equations in Hilbert Spaces via the Lacunae Method. \textit{arXiv}, arXiv:2202.07338 [math.FA],   
  Lidskii V.B.  Summability of series in terms of the principal vectors of non-selfadjoint operators. \textit{Tr. Mosk. Mat. Obs.}, \textbf{11},  (1962), 3-35. I think  the detailed description  of potential benefits of the offered method, in comparison with the ones cited above, is necessary. Here, I should note that  if we consider a fractional power of  the m-accretive  operator (in particular selfadjoint elliptic operator semibounded from bellow), then the cited above methods suit to the problem a great deal due to  the accretive property. The latter guaranties uniqueness of the solution and the  absence of any conditions imposed upon  the function (element) in the initial condition (5). Thus,  we can  apply the methods allowing to present the solution as a series for   a rather wide class of functions at the right-hand side. It is clear that   this fact gives us the opportunity to approximate the solution at least. As for other remarks, I should show my disagreement with the idea expressed on the  page 1 line 15 "... the fractional power of an elliptic operator can be defined in non unique way..." for if we consider approach by T. Kato   (Kato T.  Perturbation theory for linear operators.
   \textit{Springer-Verlag Berlin, Heidelberg, New York}, 1980.) or Krasnoselskii M.A. (Krasnoselskii M.A.,  Zabreiko P.P.,  Pustylnik E.I.,  Sobolevskii P.E.
  Integral operators in the spaces of summable functions. \textit{ Moscow:  Science,   FIZMATLIT},   1966.), then we see that the  definitions coincide with the {\it spectral definition} (as the authors called it)   (3)  by virtue of    the well-known spectral theorem. I think that the authors ought to  clarify what did they mean. Some technical remarks:\\

 

 

 

Page 1, line 2. "...parabolic problems with fractional power elliptic operators..." must be ...with  fractional powers of the elliptic operator .  \\

 I think that the authors' language abilities allow them to make a careful proofreading of the paper  by themselves.

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Sincerely yours reviewer!

Comments for author File: Comments.pdf

Author Response

Thank You for your remarks.

Author Response File: Author Response.pdf

Reviewer 3 Report

In this paper, Authors presented efficient parallel finite difference schemes for computing the solutions of 3D parabolic problems with fractional power elliptic operators and nonlinear source terms. These algorithms are targeted to solve problems with general elliptic operators in bounded domains. Discrete schemes include approximations of nonlocal operators by using the AAA algorithm. Three different approaches are used to construct efficient implicit in time schemes. The stability of schemes and convergence
rates of the discrete solution are proved. Parallel solvers are required to solve the obtained systems of linear equations. The detailed scalability analysis is done in order to compare the efficiency of different parallel algorithms. Results of computational experiments are presented and analyzed.

The paper is interesting, well written and can be accepted for publication.

Author Response

Thank You for your remarks and evaluation of our results.

Author Response File: Author Response.pdf

Round 2

Reviewer 2 Report

The paper has been significantly  improved, the comparison analysis regarding the previously obtaained  results has been implemented. I have come to the conclusion that the paper satisfies the high level  in its present form  and can be publisheed  as it is.