Multiscale modeling for contact problem with high-contrast heterogeneous coefficients with primary-dual formulation

In this paper, we propose a novel iterative multiscale framework for solving high-contrast contact problems of Signorini type. The method integrates the constrained energy minimizing generalized multiscale finite element method (CEM-GMsFEM) with a primal-dual active set strategy derived from semismooth Newton methods. First, local spectral problems are employed to construct an auxiliary multiscale space, from which energy minimizing multiscale basis functions are derived on oversampled domains, yielding a contrast-robust reduced-order approximation of the underlying partial differential equation. The multiscale bases are updated iteratively, but only at contact boundary, during the active set evolution process. Rigorous analysis is provided to establish error estimates and finite step convergence of the iterative scheme. Numerical experiments on heterogeneous media with high-contrast coefficients demonstrate that the proposed approach is both robust and efficient in capturing fine-scale features near contact boundaries.