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Published December 1, 2014 | Submitted
Journal Article Open

A parallel fast multipole method for elliptic difference equations

Abstract

A new fast multipole formulation for solving elliptic difference equations on unbounded domains and its parallel implementation are presented. These difference equations can arise directly in the description of physical systems, e.g. crystal structures, or indirectly through the discretization of PDEs. In the analog to solving continuous inhomogeneous differential equations using Green's functions, the proposed method uses the fundamental solution of the discrete operator on an infinite grid, or lattice Green's function. Fast solutions O(N)O(N) are achieved by using a kernel-independent interpolation-based fast multipole method. Unlike other fast multipole algorithms, our approach exploits the regularity of the underlying Cartesian grid and the efficiency of FFTs to reduce the computation time. Our parallel implementation allows communications and computations to be overlapped and requires minimal global synchronization. The accuracy, efficiency, and parallel performance of the method are demonstrated through numerical experiments on the discrete 3D Poisson equation.

Additional Information

© 2014 Elsevier Inc. Received 18 September 2013; Received in revised form 3 July 2014; Accepted 29 July 2014; Available online 12 August 2014.

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August 22, 2023
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October 18, 2023