Referee reports # 2, CMAME Sep. 2016 (decision: reject)

[labarba]

Revised paper submitted to CMAME on 15 March 2016 — Decision: Reject

Editor comments

I regret to inform you that the reviewers of your manuscript have advised against publication, and I must therefore decline it. For your guidance, the reviewers' comments are included below.

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Referee # 2 comments

I do not recommend this paper for publication. In this paper the authors claim to use a reduced degree of a fast multipole expansion within the GMRES iterative solution of linear algebraic systems steeming from collocation boundary element discretizations for the Laplace equation, and for the Stokes system, respectively. The reasoning and explainations are rather general and not convincing. In fact, some of the statements are not correct. See the PDF comments for a few topics of criticism.

Transcript of comments in PDF attachment

In this paper the authors claim to use a reduced degree of a fast multipole expansion within the GMRES iterative solution of linear algebraic systems steeming from collocation boundary element discretizations for the Laplace equation, and for the Stokes system, respectively. The reasoning and explainations are rather general and not convincing. In fact, some of the statements are not correct.
The discretization is done by using the first boundary integral equation with the single and double layer potential, and using piecewise constant functions to approximate both the potential and the flux. Using midpoint collocation, the linear system to be solved is obtained by using an appropriate reordering.

• 1. In equation (2), the sums do not cover the diagonal entries i = j. While for the Laplace double layer potential this is indeed true in this particular case, this is obviously not correct for the single layer potential.
• 2. In formulae (4) and (5), the factor 1/ 4π is missing.
• 3. In the second line after equation (6) it is said that x and x0 are two distinct points in the domain, which is obviously wrong since x0 has to be considered on the boundary.
• 4. Boundary conditions are discussed in a superficial way. In the case of Neu- mann boundary conditions both problems for the Laplace equation and for the Stokes system are not solvable in general, certain solvability conditions have to be assumed, and the solution is not unique. The same holds true for the Dirichlet problem for the Stokes system. These issues are neither mentioned nor discussed.
• 5. It is well known that the Dirichlet datum, i.e. the potential, and the Neumann datum, i.e. the flux, have to be approximated by using basis functions of different polynomial degree, for example piecewise linears and piecewise constants. In fact, using piecewise constants for both results in a reduced order of convergence, in particular there is no convergence when measur- ing the error in L2. So I doubt the results as given in Fig. 5, even when approximating constant functions which can be represented exactly when using piecewise constant basis functions. The shown error is then probably due to the boundary approximation of the spherical surface?
• 6. Several times it is mentioned that the optimal complexity is O(N) when using the fast multipole method. But this does not consider the Galerkin boundary element error estimate to be guaranteed.

Note that these are only a few topics of criticism.

[labarba]

Addressing Referee Comment # 1

In equation (2), the sums do not cover the diagonal entries i = j. While for the Laplace double layer potential this is indeed true in this particular case, this is obviously not correct for the single layer potential.

This is a typo on the equation.

This derivation comes from Brebbia (2nd ed., p.52, Eq. 2.26), where it is written for a particular point i. On our manuscript, simply deleting i ≠ j from the two sums will fix this and make the equation correct. The code, however, has the correct expression.

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Ref.
– Brebbia, Carlos Alberto, and Jose Dominguez. Boundary elements: an introductory course. WIT press, 2nd ed. 1992.

[labarba]

Addressing Referee Comment # 2

In formulae (4) and (5), the factor 1/ 4π is missing.

This is a typo on the equation.

Adding 1/ 4π to the right-hand side of both equations fixes this. However, the code is correct.

[labarba]

Addressing Referee Comment # 3

In the second line after equation (6) it is said that x and x0 are two distinct points in the domain, which is obviously wrong since x0 has to be considered on the boundary.

This is a terminology issue. The "domain" of integration is the boundary, as the previous passage to the line in question makes clear. We can make the sentence more clear by simply changing the word "domain" by the word "boundary."

[labarba]

Addressing Referee Comment # 4

Boundary conditions are discussed in a superficial way. In the case of Neu- mann boundary conditions both problems for the Laplace equation and for the Stokes system are not solvable in general, certain solvability conditions have to be assumed, and the solution is not unique. The same holds true for the Dirichlet problem for the Stokes system. These issues are neither mentioned nor discussed.

In problems with pure Neumann BCs the solution is non-unique (to an additive constant). This would pose an issue if we were solving interior problems, but all our problems are exterior, which implicitly places a DIrichlet BC at infinity that phi and dphi/dn are zero, and there is a unique solution.

From the classic work by Hess and Smith (1967): "…the solution of the exterior problem exists and is unique, and no difficulties are encountered in solving the equation. For the interior problem the integral is indeterminative…" (p.17)

In any case, ours is not a paper making contributions to the BEM itself, or to its analysis. Mentioning or discussing these details is inessential for our purposes. Whatever concern the referee has with these technicalities apply to any work with BEM, and using BEM on Laplace or Stokes problems is standard.

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Ref.
– Hess, John L., and A_M O. Smith. "Calculation of potential flow about arbitrary bodies." Progress in Aerospace Sciences 8 (1967): 1-138.

[labarba]

Addressing Referee Comment # 5

It is well known that the Dirichlet datum, i.e. the potential, and the Neumann datum, i.e. the flux, have to be approximated by using basis functions of different polynomial degree, for example piecewise linears and piecewise constants. In fact, using piecewise constants for both results in a reduced order of convergence, in particular there is no convergence when measur- ing the error in L2. So I doubt the results as given in Fig. 5, even when approximating constant functions which can be represented exactly when using piecewise constant basis functions. The shown error is then probably due to the boundary approximation of the spherical surface?

Although there is some consistency requirements on the order of the panels (constant panels, linear, quadratic, etc) when using Dirichlet and Neumann data, this consistency issue only applies when mixing these on the same problem. We don't have this situation. We have either a pure Dirichlet or a pure Neumann problem.

Figure 5, then, shows the spatial convergence of the boundary element solution with 1st-kind integral (Dirichlet, solid line) or 2nd-kind integral (Neumann, dotted line) separately. The two (as the figure shows) converge at different rates. This difference shows, in practice, what the referee brings up: when mixing Dirichlet and Neumann conditions in one BEM solution, the panel distributions can be selected of different order to make the convergence rate consistent. However, if one does not take this measure, the only consequence is that the overall solution will show the slowest spatial order of convergence.

[labarba]

Addressing Referee Comment # 6

Several times it is mentioned that the optimal complexity is O(N) when using the fast multipole method. But this does not consider the Galerkin boundary element error estimate to be guaranteed.

This referee comment seems to be mixing two separate things: (1) the computational complexity of the fast multipole method, which is O(N) in the sense of the time-to-solution growing linearly with the number of points; and (2) the error convergence of BEM, which goes as O(N) for Galerkin methods, in the sense of the error decreasing linearly with the number of panels.

It is absolutely correct to say that the fast multipole method has O(N) computational complexity. This was the breakthrough of Greengard and Rokhlin (1987).

Moreover, we are not using Galerkin BEM; we are using collocation.