A $p$-adaptive treecode solution of the Poisson equation in the general domain

TL;DR

This paper introduces a $p$-adaptive treecode algorithm for solving the Poisson equation on general domains, improving efficiency by adaptively adjusting the multipole expansion order based on error analysis and hierarchical geometry structures.

Contribution

It presents a novel $p$-adaptive implementation of the treecode algorithm that enhances efficiency for general domain problems by non-uniform multipole expansion orders.

Findings

Significant reduction in computational complexity.

Validated effectiveness through numerical experiments.

Achieves competitive performance in micromagnetics applications.

Abstract

Raising the order of the multipole expansion is a feasible approach for improving the accuracy of the treecode algorithm. However, a uniform order for the expansion would result in the inefficiency of the implementation, especially when the kernel function is singular. In this paper, a $p$-adaptive treecode algorithm is designed to resolve the efficiency issue for problems defined on a general domain. Such a $p$-adaptive implementation is realized through i). conducting a systematical error analysis for the treecode algorithm, ii). designing a strategy for a non-uniform distribution of the order of multipole expansion towards a given error tolerance, and iii). employing a hierarchy geometry tree structure for coding the algorithm. The proposed $p$-adaptive treecode algorithm is validated by a number of numerical experiments, from which the desired performance is observed successfully, i.e., the computational complexity is reduced dramatically compared with the uniform order case, making our algorithm a competitive one for bottleneck problems such as the demagnetizing field calculation in computational micromagnetics.