An adaptive multi-resolution framework for the meshless local PetrovGalerkin method

Abstract

This paper presents an adaptive multi-resolution framework for the Meshless Local Petrov–Galerkin (MLPG) method applied to incompressible Navier–Stokes equations. The primary contributions are: (1) a hierarchical quadtree/octree resolution management system with rigorous inter-level communication preserving partition of unity and polynomial consistency; (2) a composite error estimator integrating velocity gradients, vorticity, and residuals with proven reliability bounds; (3) conservation-preserving interface treatment via Lagrange multipliers satisfying the inf-sup condition; and (4) comprehensive convergence analysis for problems with limited regularity. Numerical experiments demonstrate computational efficiency improvements up to 4.6× compared to uniform MLPG discretizations while maintaining convergence rates approaching O(h2.1), validated on canonical benchmarks including flows with moving boundaries, geometric singularities, multi-fluid interfaces, and decaying vortices.