Hossein Hosseinzadeh

While various Meshless Local Petrov-Galerkin (MLPG) method variants exist, primarily distinguished by their choice of test functions in local subdomains, the mathematical relationships among these approaches remain unexplored. This paper establishes rigorous connections between existing MLPG formulations and proposes novel extensions based on an analysis of test function smoothness. It is shown that MLPG5, which employs Heaviside step test functions, corresponds to the mean value of MLPG2 (the collocation method), while MLPG4, using logarithmic test functions, is proven equivalent to the mean value of MLPG5 over the radius of local subdomains. Building on these insights, a systematic framework for generating new MLPG variants is introduced by leveraging the smoothness properties of test functions. All existing and newly proposed MLPG variants are demonstrated to have local weak forms equivalent to the original strong-form equations. This equivalence establishes the unique solvability of the methods, addressing long-standing questions regarding their consistency. Comprehensive numerical experiments validate the theoretical findings, confirming both the inter-variant relationships and the robustness of the newly extended MLPG variants.