Abstract:In this paper, we devise a $\operatorname{prox}$-based semi-smooth Newton method that is applicable to a finite element discretization of a broad class of nonsmooth convex variational problems, including the TV-minimization problem, the $p$-Dirichlet problem, the obstacle problem, and the elasto-plastic torsion problem. To this end, on the basis of the proximity operator, the discrete primal-dual optimality conditions are reformulated as nonlinear operator equations with Newton-differentiable structure. Under suitable assumptions on the energy densities, we establish the global well-posedness and local super-linear convergence of the resulting semi-smooth Newton method. The proposed approach coincides with established semi-smooth Newton methods for obstacle-type problems, satisfies a primal-dual invariance, and, under suitable additional assumptions, is globally well-posed in the infinite-dimensional setting.
From: Alex Kaltenbach [view email]
[v1]
Wed, 24 Jun 2026 15:23:54 UTC (1,465 KB)
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