Dirichlet-Neumann Learning Algorithm for Solving Elliptic Interface Problems

Abstract

Non-overlapping domain decomposition methods are well-suited for addressing interface problems across various disciplines, where traditional numerical simulations often require the use of interface-fitted meshes or technically designed basis functions. To remove the burden of mesh generation and to effectively tackle with the flux transmission condition, a novel mesh-free scheme, i.e., the Dirichlet-Neumann learning algorithm, is studied in this work for solving the benchmark elliptic interface problems with high-contrast coefficients and irregular interfaces. By resorting to the variational principle, we carry out a rigorous error analysis to evaluate the discrepancy caused by the boundary penalty treatment for each decomposed subproblem, which paves the way for realizing the Dirichlet-Neumann algorithm using neural network extension operators. Through experimental validation on a series of testing problems in two and three dimensions, our methods demonstrate superior performance over other alternatives even in scenarios with inaccurate flux predictions at the interface.