The Random Feature Method for Solving Interface Problems

TL;DR

This paper introduces a novel mesh-free random feature method for interface problems that avoids mesh generation, handles low regularity solutions effectively, and achieves high accuracy with significantly fewer degrees of freedom.

Contribution

The study develops a new random feature-based approach that couples solutions on either side of an interface using partition of unity, improving efficiency and robustness over traditional methods.

Findings

Achieves high accuracy comparable to spectral collocation for smooth solutions.

Requires two to three orders of magnitude fewer degrees of freedom than traditional methods.

Successfully handles solutions with low regularity, including discontinuities.

Abstract

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.