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Related Experiment Videos

A fast multipole method for Maxwell equations stable at all frequencies.

Eric Darve1, Pascal Havé

  • 1Mechanical Engineering Department, Stanford University, Stanford, CA 94305-4040, USA. darve@stanford.edu

Philosophical Transactions. Series A, Mathematical, Physical, and Engineering Sciences
|August 13, 2004
PubMed
Summary

Solving Helmholtz and Maxwell equations efficiently is crucial. We introduce a stable-plane-wave expansion method, offering lower computational cost and improved accuracy over existing fast multipole methods for these complex physics problems.

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Area of Science:

  • Computational electromagnetics
  • Numerical analysis
  • Mathematical physics

Background:

  • Integral formulations of Helmholtz and Maxwell equations result in large, dense linear systems.
  • Direct solvers are computationally expensive (O(N^3)).
  • Iterative solvers reduce cost to matrix-vector products, but require efficient computation.

Purpose of the Study:

  • To develop a more efficient and stable numerical method for solving Helmholtz and Maxwell equations.
  • To address limitations of existing fast multipole methods (FMM) in terms of accuracy and stability.

Main Methods:

  • Application of integral formulations for Helmholtz and Maxwell equations.
  • Comparison of direct solvers, iterative solvers, and FMM variants (multipole and plane-wave expansions).

Related Experiment Videos

  • Introduction and analysis of a novel 'stable-plane-wave expansion' method.

    • Main Results:

    • The proposed stable-plane-wave expansion achieves a computational complexity of O(N log N).
    • This method offers lower computational expense compared to the multipole expansion.
    • It overcomes the accuracy and stability issues associated with the standard plane-wave expansion.

    Conclusions:

    • The stable-plane-wave expansion is a computationally efficient and numerically stable approach for integral solutions of Helmholtz and Maxwell equations.
    • This method presents a significant improvement over existing FMM-based techniques for electromagnetic and wave propagation problems.