Wavelet Multi-Resolution Interpolation Galerkin Method for Linear Singularly Perturbed Boundary Value Problems

In this study, a wavelet multi-resolution interpolation Galerkin method (WMIGM) is proposed to solve linear singularly perturbed boundary value problems. Unlike conventional wavelet schemes, the proposed algorithm can be readily extended to special node generation techniques, such as the Shishkin node. Such a wavelet method allows a high degree of local refinement of the nodal distribution to efficiently capture localized steep gradients. All the shape functions possess the Kronecker delta property, making the imposition of boundary conditions as easy as that in the finite element method. Four numerical examples are studied to demonstrate the validity and accuracy of the proposed wavelet method. The results show that the use of modified Shishkin nodes can significantly reduce numerical oscillation near the boundary layer. Compared with many other methods, the proposed method possesses satisfactory accuracy and efficiency. The theoretical and numerical results demonstrate that the order of the s-uniform convergence of this wavelet method can reach 5.