Dirichlet-Neumann Learning Algorithm for Solving Elliptic Interface Problems (Communications in Computational Physics, 2025)
Non-overlapping domain decomposition methods are well-suited for addressing interface problems across various disciplines, where traditional numericalsimulations often require the use of interface-fitted meshes or technically designed basis functions. To remove the burden of mesh generation and to effectively tackle withthe flux transmission condition, a novel mesh-free scheme, i.e., the Dirichlet-Neumannlearning algorithm, is studied in this work for solving the benchmark elliptic interfaceproblems with high-contrast coefficients and irregular interfaces. By resorting to thevariational principle, we carry out a rigorous error analysis to evaluate the discrepancycaused by the boundary penalty treatment for each decomposed subproblem, whichpaves the way for realizing the Dirichlet-Neumann algorithm using neural networkextension operators. Through experimental validation on a series of testing problemsin two and three dimensions, our methods demonstrate superior performance overother alternatives even in scenarios with inaccurate flux predictions at the interface.