Abstract
Fast stray field calculation is commonly considered of great importance for micromagnetic simulations, since it is the most time consuming part of the simulation. The Fast Multipole Method (FMM) has displayed linear O(N) parallelization behavior on many cores. This article investigates the error of a recent FMM approach approximating sources using linear—instead of constant—finite elements in the singular integral for calculating the stray field and the corresponding potential. After measuring performance in an earlier manuscript, this manuscript investigates the convergence of the relative L2 error for several FMM simulation parameters. Various scenarios either calculating the stray field directly or via potential are discussed.
Highlights
Every distributed computational solution to the laplace equation—in this case the stray field— has to deal with its bad scalability properties
To achieve its superior performance the Fast Multipole Method (FMM) approximates the discretized problem by using multipole and local expansions
The direct stray field solution of potential or field is required for a whole class of algorithms including FMM, non uniform grid, fast Fourier transform and the tensor grid method, making this publication a reference for the implementation of all of the mentioned algorithms.[2]
Summary
Every distributed computational solution to the laplace equation—in this case the stray field— has to deal with its bad scalability properties. The fast multipole method has been shown to exhibit high parallel efficiency (> 70%) across thousands of cores in memory and time1—albeit for a point like Laplace kernel. While an upper limit for the error can be theoretically computed no actual error data has been published, making it hard to determine applicability of FMM, choosing a suiting implementation and verifying the correct implementation. The direct stray field solution of potential or field is required for a whole class of algorithms including FMM, non uniform grid, fast Fourier transform and the tensor grid method, making this publication a reference for the implementation of all of the mentioned algorithms.[2]
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