Recovery of functions from weak data using unsymmetric meshless kernel-based methods

Recent engineering applications successfully introduced unsymmetric meshless local Petrov–Galerkin (MLPG) schemes. As a step towards their mathematical analysis, this paper investigates nonstationary unsymmetric Petrov–Galerkin-type meshless kernel-based methods for the recovery of L2 functions from finitely many weak data. The results cover solvability conditions and error bounds in negative Sobolev norms with partially optimal rates. These rates are mainly determined by the approximation properties of the trial space, while choosing sufficiently many test functions ensures stability. Numerical examples are provided, supporting the theoretical results and leading to new questions for future research.