While linear FETI-DP (Finite Element Tearing and Interconnecting - Dual Primal) is an efficient iterative domain decomposition solver for discretized linear PDEs (partial differential equations), nonlinear FETI-DP is its consequent extension to the nonlinear case. In both methods, the parallel efficiency of the method results from a decomposition of the computational domain into nonoverlapping subdomains and a resulting localization of the computational work. For a fast linear convergence of the linear FETI-DP method, a global coarse problem has to be considered. Adaptive coarse spaces are provably robust variants for many complicated micro-heterogeneous problems, as, for example, stationary diffusion problems with large jumps in the diffusion coefficient. Unfortunately, the set-up and exact computation of adaptive coarse spaces is known to be computationally expensive. Therefore, recently, surrogate models based on neural networks have been trained to directly predict the adaptive coarse constraints. Here, these learned constraints are implemented in nonlinear FETI-DP and it is shown numerically that they are able to improve the nonlinear as well as linear convergence speed of nonlinear FETI-DP.

翻译：线性FETI-DP（有限元撕裂与互联-对偶原始）是求解离散化线性偏微分方程的高效迭代区域分解求解器，而非线性FETI-DP则是将其自然地扩展至非线性情形。在这两种方法中，并行效率源于将计算域分解为互不重叠的子域，从而实现计算工作量的局部化。为加速线性FETI-DP方法的线性收敛速度，需引入全局粗问题。自适应粗空间对于许多复杂微异质问题（例如扩散系数存在大幅跳跃的稳态扩散问题）具有可证明的鲁棒性。然而，构建并精确计算自适应粗空间在计算上极为昂贵。为此，近年来研究者训练基于神经网络的代理模型以直接预测自适应粗约束。本文将此类学习约束应用于非线性FETI-DP方法，数值实验表明，其能有效提升非线性FETI-DP方法的非线性与线性收敛速度。