TMP2D: rank deficiency of multi-centre harmonic expansions and a well-conditioned local Taylor algorithm — software and reproducibility package

Résumé fourni par la source

Software accompanying the article Rank deficiency of multi-centre harmonic expansions and a well-conditioned local Taylor algorithm for planar elliptic problems by D. S. Zézé, Y. Kramoh and F. N'doli Koffi, submitted to Numerical Algorithms (2026). The negative result. The harmonic polynomials of degree at most p form a translation-invariant space of dimension 2p+1. Summing harmonic expansions about n distinct centres therefore spans 2p+1 dimensions whatever n may be, and the collocation matrix carries an exact rank defect of (n-1)(2p+1). The archive retains the earlier single-global-basis implementation (TMP2D/solver.py, TMP2D/solver_extended.py) precisely so that this failure and the reported condition numbers can be reproduced. The algorithm. The domain is partitioned; on each block a truncated Taylor expansion is condensed by the equation, leaving 2p+1 free coefficients; blocks are coupled by continuity of the trace and of the normal flux, the system is closed by boundary collocation and solved in the least-squares sense through a truncated SVD. Rectangles, L-shaped domains, disks and annuli are supported, the last through a basis enriched with log r and negative powers; polynomial sources are handled by particular solutions. Reproducibility. python3 reproduce_all.py regenerates every figure and every table of the article in a few minutes, writing a full transcript to results/reproduce_all.log. No value reported in the article is transcribed by hand. Version 1.1 adds study_sampling.py, which measures the sampling threshold of Section 7 (a collocation-to-unknown ratio near 1.5 separates a regime with no correct digit in double precision from one with condition numbers of order ten), and figlabels.py, a language switch producing figures with English or French labels. Requires Python 3.9+, NumPy, SciPy and Matplotlib. Nothing is compiled; no external data file and no network access are required.