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<p>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<InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\times \)</EquationSource> </InlineEquation> compared to uniform MLPG discretizations while maintaining convergence rates approaching <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\mathcal {O}(h^{2.1})\)</EquationSource> </InlineEquation>, validated on canonical benchmarks including flows with moving boundaries, geometric singularities, multi-fluid interfaces, and decaying vortices.</p>

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An adaptive multi-resolution framework for the meshless local Petrov-Galerkin method

  • 期刊论文

  • 发表日期：

    2025年12月1日

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  • Amirkeivan Shafiei,
  • Seyed Mojtaba Mosavi Nezhad

摘要

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 \(\times \) compared to uniform MLPG discretizations while maintaining convergence rates approaching \(\mathcal {O}(h^{2.1})\) , validated on canonical benchmarks including flows with moving boundaries, geometric singularities, multi-fluid interfaces, and decaying vortices.