TY - JOUR
T1 - Dirichlet-Neumann Learning Algorithm for Solving Elliptic Interface Problems
AU - Sun , Qi
AU - Xu , Xuejun
AU - Yi , Haotian
JO - Communications in Computational Physics
VL - 1
SP - 248
EP - 284
PY - 2025
DA - 2025/07
SN - 38
DO - http://doi.org/10.4208/cicp.OA-2024-0046
UR - https://global-sci.org/intro/article_detail/cicp/24258.html
KW - Elliptic interface problem, high-contrast coefficient, compensated deep Ritz method, artificial neural networks, deep learning.
AB - <p style="text-align: justify;">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.</p>