Available online at www.sciencedirect.com ScienceDirect Mathematics and Computers in Simulation 180 (2021) 379–400 www.elsevier.com/locate/matcom Original articles A stable computation on local boundary-domain integral method for elliptic PDEs L. Ponzellini Marinelli ∗ , N. Caruso, M. Portapila National University of Rosario - Faculty of Exact Sciences, Engineering and Surveying, Rosario S2000BTP, Argentina CIFASIS, French Argentine International Center for Information and Systems Sciences, UAM (France)/UNR-CONICET (Argentina), S2000EZP Rosario, Argentina Received 30 October 2019; received in revised form 29 August 2020; accepted 31 August 2020 Available online 8 September 2020 Abstract Many local integral methods are based on an integral formulation over small and heavily overlapping stencils with local Radial Basis Functions (RBFs) interpolations. These functions have become an extremely effective tool for interpolation on scattered node sets, however the ill-conditioning of the interpolation matrix – when the RBF shape parameter tends to zero corresponding to best accuracy – is a major drawback. Several stabilizing methods have been developed to deal with this near flat RBFs in global approaches but there are not many applications to local integral methods. In this paper we present a new method called Stabilized Local Boundary Domain Integral Method (LBDIM-St) with a stable calculation of the local RBF approximation for small shape parameter that stabilizes the numerical error. We present accuracy results for some Partial Differential Equations (PDEs) such as Poisson, convection–diffusion, thermal boundary layer and an elliptic equation with variable coefficients. c ⃝ 2020 International Association for Mathematics and Computers in Simulation (IMACS). Published by Elsevier B.V. All rights reserved. Keywords: Local integral method; Radial basis functions; Shape parameter; ill-conditioning; RBF-QR method; PDE 1. Introduction The Boundary Element Method (BEM) is a well-established numerical technique in engineering. This method is based on the transformation of the original partial differential equation that define a given physical problem, into an equivalent integral equation by means of the corresponding Green’s second identity and its fundamental solution. When using BEM for large problems, with or without closed form fundamental solution, a domain decomposition technique is frequently used, in which the original domain is divided into subdomains, and on each of them the full integral representation formulae are applied. Meshless formulations of local BEM approaches, see Zhu et al. [27], are attractive and efficient techniques to improve the performance of local BEM schemes. In these methods, the integral representation formulae are applied at local internal integration subdomains embedded into interpolation stencils that are heavily overlapped. Different interpolation schemes can be employed at the interpolation stencils, ∗ Correspondence to: CIFASIS, French Argentine International Center for Information and Systems Sciences, UAM (France)/UNR-CONICET (Argentina), S2000EZP Rosario, Argentina. E-mail addresses: ponzellini@cifasis-conicet.gov.ar (L. Ponzellini Marinelli), caruso@cifasis-conicet.gov.ar (N. Caruso), portapila@cifasis-conicet.gov.ar (M. Portapila). https://doi.org/10.1016/j.matcom.2020.08.027 0378-4754/ c ⃝ 2020 International Association for Mathematics and Computers in Simulation (IMACS). Published by Elsevier B.V. All rights reserved.