Introduction
The Meshless Local Petrov-Galerkin (MLPG) method solves the weak form of equations locally without a background mesh, making it suitable for flows with moving boundaries or complex geometries. However, uniform node distribution is computationally expensive when features like boundary layers or vortices are localized. Recent adaptive advancements include error-estimator-guided h-adaptive MLPG and quadtree-based RBF-FD methods.
This paper introduces a dynamic adaptive multiscale MLPG algorithm with four key contributions:
Hierarchical basis functions preserving partition of unity and polynomial reproduction across resolution levels.
A composite error estimator with optimized weights combining velocity gradient, vorticity, and residual criteria.
Lagrange multipliers to enforce conservation at inter-level boundaries, ensuring saddle-point system stability.
Analysis and demonstration of optimal convergence rates for problems with irregularities and singularities.
Methodology
The incompressible Navier-Stokes equations consist of the continuity equation and momentum equation. The weak form is expressed locally over subdomain Ω_i using weight function w. The computational domain is partitioned using quadtree structure in 2D or octree in 3D into different levels, with nodal spacing at level l given by h_l=h_0⋅2^(-l).
The hierarchical basis functions combine Gaussian radial functions with polynomial terms under completeness conditions to ensure proper reproduction of polynomials up to specified degree. At inter-level boundaries, a C^1-continuous blending function is used with transition width coefficient w_T=3.0 to ensure smooth and continuous transition between levels.
Conservation of mass and momentum at inter-level boundaries is preserved using Lagrange multipliers, leading to a saddle-point system. The stability of this system is guaranteed by satisfying the inf-sup condition with constant β>0 independent of refinement level h_l.
A composite error estimator is employed to identify regions requiring refinement, combining three criteria with optimized weights:
η_i=0.4ϵ ̃_(∇u,i)+0.35ϵ ̃_(ω,i)+0.25ϵ ̃_(r,i), (1)
where the components represent normalized velocity gradient norm, vorticity norm, and equation residual respectively. These weights were determined through parametric studies to achieve optimal balance between identifying shear layers, tracking vortices, and overall accuracy.
Key parameters were determined through sensitivity analysis: transition width w_T=3.0, refinement thresholds (θ,θ_c)=(0.3,0.1), and RBF shape parameter c=1.2h_avg. Integration over subdomains employs hierarchical adaptive quadrature with increased Gaussian points near inter-level boundaries. The saddle-point system is solved using restarted GMRES with block-diagonal preconditioning, showing convergence independent of refinement level with iteration counts remaining in the range 35-75
Results and Discussion
The proposed method was tested on four standard benchmark problems.
For the moving boundary flow problem with a circular obstacle at Reynolds number 200, AMR-MLPG achieved L2 error of 3.17×10(-4) using 24,362 nodes compared to uniform MLPG with error 4.28×10(-4) using 90,000 nodes, representing approximately 78% computational savings. The observed convergence rate was O(h^2.1) , exceeding the theoretical rate and consistent with super convergence theory in adaptive FEM.
For the reentrant corner flow problem with an L-shaped domain at Reynolds number 500 featuring velocity singularity u∼r(2/3) at the corner, the measured grading ratio ρ=0.72±0.05 satisfies the theoretical condition, and mesh size scaling h(r)∼r0.41 agrees with theoretical value β=3/7≈0.43. The singularity exponent extracted from numerical results was 0.668, showing only 0.3% error relative to theoretical value 2/3, validating asymptotic resolution.
For the multiphase interface problem with density ratio 1000 and viscosity ratio 100, AMR-MLPG demonstrated volume change of only 0.18% and interface error 2.36×10(-4) using 26,518 nodes, outperforming VOF-AMR with 0.75% volume change and 42,612 nodes. Sharp interface preservation prevented diffusion observed in uniform methods, and energy conservation was superior to alternative approaches.
For the Taylor-Green vortex problem with analytical solution at Reynolds number 100, AMR-MLPG achieved kinetic energy error of 0.28% and vorticity error 3.42×10(-4) using 18,756 nodes, approaching the accuracy of spectral element methods. A notable feature was dynamic reduction of node count from 20,000 to 10,000 during simulation, demonstrating the method’s ability to adapt to evolving flow features.
The super optimal convergence rate O(h2.1) observed in moving boundary and Taylor-Green problems exceeds the theoretical O(h2) for second-order bases. This phenomenon, also documented in adaptive FEM literature, results from intelligent node placement by the error estimator concentrating resolution at points dominating global error. Efficiency improvements ranging from 3.5 to 4.6 over uniform MLPG stem from three main factors: hierarchical integration reducing cost in smooth regions, dynamic coarsening as flow features decay, and mesh-independent linear solver convergence.
Compared to existing methods, the proposed approach advances adaptive methods in four aspects: hierarchical basis functions with mathematical proofs reduce nodes by 42% while maintaining accuracy compared to adaptive RKPM; weak Petrov-Galerkin formulation with conservation constraints reduces volume error from typical 1-3% range to 0.18% compared to adaptive RBF-FD; Lagrange multiplier approach eliminates remeshing overhead for moving boundaries, reducing computational cost by 56% compared to adaptive SPH; and support domain flexibility improves corner error resolution by one order compared to adaptive h-FEM.
Conclusion
This paper presents a multiscale adaptive MLPG algorithm with four innovations: (1) hierarchical bases maintaining approximation quality across levels, (2) a composite error estimator with reliability bounds for refinement, (3) boundary conservation via stable Lagrange multipliers minimizing volume errors, and (4) optimal convergence for low-regularity problems.
Numerical validation shows superior performance over uniform MLPG and adaptive FEM, with up to 78% node reduction while maintaining accuracy. The observed O(h²˙¹) convergence and dynamic adaptation make it suitable for complex flows with localized features, moving boundaries, and singularities.