A proximal bundle algorithm for a class of quasilinear variational inequalities of the second kind arising in the viscoplastic laminar flow

This paper focuses on the numerical solution of a variational inequality of the second kind, which arises as a model for the laminar flow of a Herschel-Bulkley fluid in the cross-section of a pipe.To tackle this problem, we develop a nonsmooth proximal bundle algorithm that bypasses the need for regularization techniques.We begin by formulating and analyzing an associated nonsmooth and convex optimization problem that characterizes the solution of the variational inequality.Following a discretize-then-optimize approach, we employ a first-order finite element discretization for the objective functional.The core of our method lies in the nonsmooth bundle algorithm, which leverages a Moreau-Yosida approximation combined with a quasi-Newton BFGS update.This approach approximates the function and gradient values through a finite inner bundle algorithm.We build and analyze the proposed algorithm, examining its convergence properties in the context of the flow model.Additionally, we demonstrate its efficiency through both theoretical analysis and numerical experiments.