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insight - Numerical analysis - # Convergence and oscillations in MFS solutions of Laplace-Neumann problems

Core Concepts

The method of fundamental solutions (MFS) can generate convergent solutions to Laplace-Neumann problems even when the intermediate auxiliary source currents exhibit unphysical divergence and oscillations.

Abstract

The paper examines the convergence and oscillatory behavior of MFS solutions for two-dimensional Laplace-Neumann problems, focusing on exterior circular problems.
Key highlights:
The MFS can generate the correct solution even when the intermediate auxiliary source currents diverge and oscillate rapidly, a phenomenon that is not well-known.
This occurs when the auxiliary sources are placed behind the singularities of the analytic continuation of the scattered potential.
In the "physical" case where the auxiliary sources enclose the singularities, the normalized auxiliary currents converge to a continuous surface current density.
In the "unphysical" case where the auxiliary sources do not enclose the singularities, the auxiliary currents exhibit exponentially large oscillations, but the final MFS solution still converges to the true potential.
The condition number of the MFS system grows exponentially with the number of auxiliary sources, but the final potential is insensitive to errors in the auxiliary currents due to a low-pass filtering effect.
The main findings are extended to a noncircular Laplace-Neumann problem, demonstrating that the unphysical oscillations are a general feature of the MFS for such problems.

Stats

The paper provides the following key figures and equations:
Equation (4.15): Asymptotic formula for the MAS currents Iℓ in the "unphysical" case where the auxiliary sources do not enclose the singularities.
Equation (4.23): Exponential growth of the condition number κ of the MFS system as the number of auxiliary sources N increases.
Equation (5.10): Condition number κ′ for the MFS scheme using traditional fundamental solutions, which grows less rapidly than κ.

Quotes

"Oscillations also go unnoticed in the recent work [28], which considers the stability and errors of MFS within the context of the Laplace equation with Neumann boundary conditions."
"To the best of our knowledge, this is the first work discussing MAS-source oscillations within the specific context of Laplace-Neumann problems."
"Nonetheless, the phenomenon of MAS-source oscillations often goes unnoticed; this is true even for papers aiming to discuss the convergence of MFS/MAS, an example being the recent review article [8]."

Key Insights Distilled From

by Georgios D. ... at **arxiv.org** 04-12-2024

Deeper Inquiries

The unphysical oscillations in the MFS auxiliary currents, as discussed in the context of Laplace-Neumann problems, are related to the phenomenon of internal resonances in time-harmonic problems. Internal resonances occur when the system's natural frequency coincides with an external driving frequency, leading to large oscillations in the system's response. Similarly, in the MFS context, the oscillations in the auxiliary currents can be seen as a response to certain conditions or constraints within the problem. These oscillations may not have a physical interpretation but are a result of the numerical method trying to converge to the correct solution. Just like internal resonances, these oscillations can have a significant impact on the behavior of the system and need to be understood and managed effectively.

The insensitivity of the final MFS solution to errors in the auxiliary currents has important implications for the practical implementation of the method. This insensitivity means that small errors in the determination of the auxiliary currents, which may arise due to numerical issues like roundoff errors, do not significantly affect the final solution obtained from the MFS. This is a valuable characteristic as it provides some robustness to the method and allows for a certain level of error tolerance in the computation process. However, it is essential to note that while the final solution may be insensitive to these errors, it is still crucial to minimize errors in the calculation of auxiliary currents to ensure the overall accuracy of the MFS solution.

The insights gained from the study of Laplace-Neumann problems using the MFS can indeed be extended to other types of boundary value problems solved using the method. The understanding of unphysical oscillations, convergence properties, and the behavior of the MFS solutions in different scenarios can provide valuable guidance for applying the method to a wide range of problems. By recognizing the factors that influence the accuracy and stability of the MFS solutions, researchers and practitioners can adapt and optimize the method for various applications. The principles and findings from this study can serve as a foundation for exploring and improving the effectiveness of the MFS in solving diverse boundary value problems.

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