With this in mind, there are numerous ways to evaluate the action of K approximately, using discrete methods for approximating the solution of Poisson's equation. For example, the multigrid method can be used to do this in a very efficient way. The analogy with quadrature suggests using a grid with q points and solving Poisson's equation, with a multigrid method in O(q) work. Then the complete evaluation could be done in O(qmn) work. This potentially would be faster than direct evaluation of the integrals.

References

Bagheri, B., L.R. Scott, and S. Zhang, 1994, Implementing and using high-order finite element methods, Finite Elements in Analysis and Design 16:175–189.


Ding, Hong-Qiang, Naoki Karasawa, and William A. Goddard III, 1992, Atomic level simulations on a million particles: The cell multipole method for Coulomb and London nonbond interactions, J. Chem. Phys. 97:4309–4315.

Draghicescu, C.I., 1994, An efficient implementation of particle methods for the incompressible Euler equations, SIAM J. Numer. Anal. 31: 1090–1108.



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