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 W 2 1 . We prove that the proposed method is stable. The optimal convergence orders under H 2 , H 1 and L 2 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.

中文翻译：

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一维椭圆界面问题的四阶最小二乘再现核方法

摘要 界面问题的数值模拟因其在科学的各个方面的广泛应用而受到越来越多的关注。在增强再生核的应用的推动下，我们在本文中专注于建立破三次样条空间和开发一维椭圆界面问题的四阶数值方案。该方法基于最小二乘法和破碎三次样条空间。通过对W 2 1 的再生核函数进行积分，得到建立的破三次样条空间的一组新基。我们证明了所提出的方法是稳定的。还讨论了H 2 、H 1 和L 2 范数下的最优收敛阶数。我们的主要贡献是我们的对称方法是稳定的，并且可以自然地扩展到更高阶的方案。最后，我们的理论发现得到了几个数值实验的验证。此外，给出了所提出的方法与差电位方法、再生核方法和浸入式有限元方法的一些比较。