An Adaptive Element-Free Galerkin Approach for Solving Singularly Perturbed Boundary Layer Problems

The paper’s objective is to propose a robust and dynamic mesh-free numerical approach to solve singularly perturbed problems(SPPs). As is well recognized that solutions of SPPs yield boundary layers as the singular perturbation parameter approaches to zero and the conventional approaches fail to approximate these solutions, especially in the boundary layer region. In the present work, element-free Galerkin (EFG) approach has been proposed to capture these solutions with a high precision of accuracy. The key benefit of the suggested approach is that there is no need for mesh or element connectivity during implementation. Drive to this advantage, in the paper, non-uniformly distributed nodes have been constructed which condense in the boundary layer region. The moving least-squares (MLS) approximation has been employed to generate the shape functions. The proposed approach is based on global weak form and involves background cells for numerical integration computations. Essential boundary conditions have been enforced by the incorporation of the Lagrange multiplier method. In order to verify the computational consistency and robustness of the EFG scheme, a variety of numerical examples have been considered and \( L_\infty \) errors have been presented. Comparisons of solutions have been made with those available in the literature.