On the Accuracy and Efficiency of Galerkin and Direct Methods for Second-Order Boundary Value Problems with Neumann Boundary Conditions

Abstract

This study presents a numerical analysis of a second-order mixed boundary value problem with Neumann boundary condition. The problem is first formulated and solved using the Galerkin method, where the Neumann condition is naturally incorporated through weak formu-lation. To independently verify the Galerkin solution, a direct numerical approach based on a shooting technique combined with a DMVS adaptive integrator is employed, and the unknown initial derivative is determined using a secant iteration. A detailed comparison between the analytical solution, the Galerkin approximation, and the direct numerical results is carried out in terms of solution profiles, initial slope, and error norms. The numerical results demonstrate excellent agreement between both approaches, confirming the accuracy, stability, and reliability of the Galerkin formulation. The study highlights that while the direct method offers computa-tional efficiency, the Galerkin method provides a systematic and robust framework suitable for mixed boundary value problems.