A fourth-order least-squares based reproducing kernel method for one-dimensional elliptic interface problems

Abstract

Increased attention has been paid on numerical modeling of interface problems as its wide applications in various aspects of science. Motivated by enhancing the application of the reproducing kernel, we focus on establishing a broken cubic spline spaces and developing a fourth-order numerical scheme for one-dimensional elliptic interface problems in this paper. The method is based on the least-squares method and broken cubic spline spaces. A new set of basis of the established broken cubic spline space are obtained by integrating the reproducing kernel function of W21. We prove that the proposed method is stable. The optimal convergence orders under H2, H1 and L2 norms are also discussed. Our main contribution is that our symmetric method is stable and can be naturally extended to higher order schemes. Finally, our theoretical findings are verified by several numerical experiments. In addition, some comparisons of the proposed method with difference potentials methods, reproducing kernel methods and immersed finite element methods are given.