TY - JOUR
T1 - A $p$-Adaptive Treecode Solution of the Poisson Equation in the General Domain
AU - Cui , Zixuan
AU - Yang , Lei
JO - Communications in Computational Physics
VL - 5
SP - 1331
EP - 1354
PY - 2025
DA - 2025/09
SN - 38
DO - http://doi.org/10.4208/cicp.OA-2024-0320
UR - https://global-sci.org/intro/article_detail/cicp/24459.html
KW - Treecode, $p$-adaptive method, hierarchy geometry tree, Poisson equation.
AB - <p style="text-align: justify;">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 method 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.</p>