Interface problems have long been a major focus of scientific computing, leading to the development of various numerical methods. Traditional mesh-based methods often employ time-consuming body-fitted meshes with standard discretization schemes or unfitted meshes with tailored schemes to achieve controllable accuracy and convergence rate. Along another line, mesh-free methods bypass mesh generation but lack robustness in terms of convergence and accuracy due to the low regularity of solutions. In this study, we propose a novel method for solving interface problems within the framework of the random feature method. This approach utilizes random feature functions in conjunction with a partition of unity as approximation functions. It evaluates partial differential equations, boundary conditions, and interface conditions on collocation points in equal footing, and solves a linear least-squares system to obtain the approximate solution. To address the issue of low regularity, two sets of random feature functions are used to approximate the solution on each side of the interface, which are then coupled together via interface conditions. We validate our method through a series of increasingly complex numerical examples. Our findings show that despite the solution often being only continuous or even discontinuous, our method not only eliminates the need for mesh generation but also maintains high accuracy, akin to the spectral collocation method for smooth solutions. Remarkably, for the same accuracy requirement, our method requires two to three orders of magnitude fewer degrees of freedom than traditional methods, demonstrating its significant potential for solving interface problems with complex geometries.

翻译：界面问题长期以来一直是科学计算的主要关注点，由此催生了各种数值方法的发展。传统的基于网格的方法通常采用耗时的贴体网格配合标准离散格式，或采用非贴体网格配合定制化格式，以实现可控精度和收敛速度。另一类无网格方法虽然避免了网格生成，但由于解的低正则性，在收敛性和精度方面缺乏稳健性。在本研究中，我们提出了一种新的方法，在随机特征方法框架内求解界面问题。该方法利用随机特征函数结合单位分解作为近似函数，将偏微分方程、边界条件和界面条件在配置点上平等处理，通过求解线性最小二乘系统获得近似解。为解决低正则性问题，我们在界面两侧分别使用两组随机特征函数逼近解，并通过界面条件将它们耦合起来。我们通过一系列复杂度递增的数值算例验证了所提方法。结果表明，尽管解通常仅有连续性甚至不连续，但我们的方法不仅无需生成网格，还能保持高精度，类似于求解光滑解时的谱配置法。值得注意的是，在同等精度要求下，该方法所需的自由度比传统方法低两到三个数量级，展现了其在处理复杂几何形状界面问题中的巨大潜力。