A Natural Decomposition Method for Essential Boundary Conditions in Noninterpolatory Meshfree Spaces

Abstract

This paper develops a natural decomposition method (NDM)for imposing essential boundary conditions in noninterpolatory meshfree Galerkin spaces without boundary parameter tuning or auxiliary constraint construction. In such spaces, algebraic coefficients generally do not coincide with boundary values; hence coefficient assignment or nodal boundary prescription is not equivalent to imposing the continuous trace required by the variational problem. NDM introduces boundary data before discretization through a natural transfer mechanism: a source subproblem accounts for the forcing term, a weighted curl correction transfers the remaining trace mismatch, and a scalar recovery step reconstructs the solution from the corrected weighted gradient. For topologically trivial single domains with connected boundary, the reconstructed solution is equivalent, at the continuous level, to the solution satisfying the prescribed essential boundary data. The discrete analysis separates the approximation defect of the recovery space from the upstream transfer error visible to that space. Numerical experiments on benchmark problems evaluate the proposed transfer mechanism and report the associated conditioning, computational cost, and boundary perturbation behavior.