Domain decomposition methods are robust and parallel scalable, preconditioned iterative algorithms for the solution of the large linear systems arising in the discretization of elliptic partial di↵erential equations by finite elements. The convergence rate of these methods is generally determined by the eigenvalues of the preconditioned system. For second-order elliptic partial di↵erential equations, coe cient discontinuities with a large contrast can lead to a deterioration of the convergence rate. A remedy can be obtained by enhancing the coarse space with elements, which are often called constraints, that are computed by solving small eigenvalue problems on portions of the interface of the domain decomposition, i.e., edges in two dimensions or faces and edges in three dimensions. In the present work, without restriction of generality, the focus is on two dimensions. In general, it is di cult to predict where these constraints have to be added and therefore the corresponding local eigenvalue problems have to be computed, i.e., on which edges. Here, a machine learning based strategy using neural networks is suggested to predict the geometric location of these edges in a preprocessing step. This reduces the number of eigenvalue problems that have to be solved before the iteration. Numerical experiments for model problems and realistic microsections using regular decompositions as well as decompositions from graph partitioners are provided, showing very promising results.

New nonlinear FETI-DP (dual-primal finite element tearing and interconnecting) and BDDC (balancing domain decomposition by constraints) domain decomposition methods are introduced. In all these methods, in each iteration, local nonlinear problems are solved on the subdomains. The new approaches can significantly reduce communication and show a significantly improved performance, especially for problems with localized nonlinearities, compared to a standard Newton-Krylov-FETI-DP or BDDC approach. Moreover, the coarse space of the nonlinear FETI-DP methods can be used to accelerate the Newton convergence. It is also found that the new nonlinear FETI-DP and nonlinear BDDC methods are not as closely related as in the linear context. Numerical results for the p-Laplace operator are presented.

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