The interface problems are faced with multiple connected domains and consequently, their solutions or derivatives might be discontinuous. This paper proposes the use of collocation based radial basis function partition of unity method (RBF-PUM) for solving two-dimensional elliptic interface problems. The RBF-PUM is a local method that allows overcoming the high computational cost associated with the global RBF methods. In the RBF-PUM, the domain is split into overlapping patches forming a covering of it. However, this method suffers from instability when the RBF shape parameter $\epsilon$ tends to zero. To overcome this issue, we use the RBF-QR algorithm which offers stable computations for all values of $\epsilon$ and provides higher accuracy. To obtain the appropriate solution in the vicinity of the interface, the domain decomposition technique is used. In this technique, the approximation in each subdomain is built separately, and proper jump conditions are then imposed across the interface. We illustrate how to apply the proposed method to Sturm-Liouville, Sturm-Liouville eigenvalue and elastostatic interface problems. The proposed method in dealing with arbitrary interfaces within different domain sizes is validated. We present some numerical examples in which the results are compared with exact solutions and those provided by other numerical methods.

接口问题面临多个连接域，因此，它们的解决方案或衍生产品可能是不连续的。本文提出使用统一方法的基于径向的径向基函数划分（RBF-PUM）来解决二维椭圆界面问题。RBF-PUM是一种本地方法，可以克服与全局RBF方法相关的高计算成本。在RBF-PUM中，将域拆分为重叠的补丁，以覆盖它。但是，当RBF形状参数出现时，该方法会出现不稳定的情况$\epsilon$趋于零。为解决此问题，我们使用RBF-QR算法，该算法可针对所有$\epsilon$并提供更高的准确性。为了在界面附近获得合适的解决方案，使用了域分解技术。在此技术中，每个子域中的近似值是分别构建的，然后在接口上施加适当的跳转条件。我们说明了如何将建议的方法应用于Sturm-Liouville，Sturm-Liouville特征值和弹性界面问题。验证了所提出的处理不同域大小内的任意接口的方法。我们提供一些数值示例，在其中将结果与精确解进行比较，并通过其他数值方法提供这些结果。