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Fast Spectral Collocation Method for Surface Integral Equations of Potential Problems in a Spheroid.

Zhenli Xu¹, Wei Cai

¹Department of Mathematics and Statistics, University of North Carolina at Charlotte, Charlotte, NC 28223, USA.

Communications in Computational Physics

|September 28, 2011

Summary

This study introduces a faster spectral collocation method for spheroid potential problems. The new technique significantly reduces computation costs for spectral expansion calculations.

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Area of Science:

Computational Mathematics
Numerical Analysis
Potential Theory

Background:

Solving potential problems over spheroids often involves computationally intensive surface integral equations.
Spectral collocation methods offer high accuracy but can be limited by computational cost, particularly for large matrices.
Efficient computation of spectral collocation matrices is crucial for advancing numerical solutions in potential theory.

Purpose of the Study:

To develop and present a novel technique for accelerating the computation of spectral collocation discretizations for surface operators over spheroids.
To reduce the computational complexity associated with solving potential problems using spectral methods on spheroidal geometries.

Main Methods:

Approximation of layer densities using spectral expansion of spherical harmonics.
Application of the spectral collocation method to solve surface integral equations.
Development of a new technique to optimize the calculation of collocation matrix entries.

Main Results:

The proposed technique reduces the computation cost of collocation matrix entries from O(M^2 N^4) to O(MN^4).
This represents a significant speed-up in the matrix computation, where N is the number of spherical harmonics and M is the number of quadrature points.
Numerical results confirm the spectral accuracy of the developed method.

Conclusions:

The new technique effectively accelerates the computation of spectral collocation discretizations for surface operators over spheroids.
This advancement offers a more efficient approach to solving potential problems in spheroidal domains.
The method maintains spectral accuracy, making it a valuable tool for computational mathematics and physics.