Critique on "Volume penalization for inhomogeneous Neumann boundary conditions modeling scalar flux in complicated geometry"

TL;DR

This paper critically examines the accuracy of Sakurai et al.'s volume penalization method for inhomogeneous Neumann boundary conditions, providing counter-examples and demonstrating that its convergence rate varies between first and second order depending on the problem.

Contribution

The paper offers new counter-examples showing that Sakurai et al.'s method does not always achieve second-order accuracy and highlights the need for further analysis of its convergence properties.

Findings

Accuracy varies between O(1) and O(2) depending on the problem.

Reproducibility issues with previous results are demonstrated.

Spatial convergence rate cannot be predicted solely based on interface shape or flux values.

Abstract

In this letter, we provide counter-examples to demonstrate that it is possible to retain second-order accuracy using Sakurai et al.'s method, even when different flux boundary conditions are imposed on multiple interfaces that do not conform to the Cartesian grid. We consider both continuous and discontinuous indicator functions in our test problems. Both indicator functions yield a similar convergence rate for the problems considered here. We also find that the order of accuracy results for some of the cases presented in Sakurai et al. are not reproducible. This is demonstrated by re-considering the same one- and two-dimensional Poisson problems solved in Sakurai et al. in this letter. The results shown in this letter demonstrate that the spatial order of accuracy of the flux-based VP approach of Sakurai et al. is between $\mathcal{O}$(1) and $\mathcal{O}$(2), and it depends on the underlying problem/model. The spatial order of accuracy cannot simply be deduced a priori based on the imposed flux values, shapes, or grid-conformity of the interfaces, as concluded in Sakurai et al. Further analysis is required to understand the spatial convergence rate of the flux-based VP method.