A Preconditioned Descent Algorithm for Variational Inequalities of the Second Kind Involving the $p$-Laplacian Operator

TL;DR

This paper introduces a preconditioned descent algorithm for solving variational inequalities involving the p-Laplacian, with applications to non-Newtonian fluid modeling, demonstrating convergence and efficiency through numerical experiments.

Contribution

It develops a novel preconditioned descent method for second-kind variational inequalities with p-Laplacian, including convergence analysis and practical numerical implementation.

Findings

The regularized problems are well-posed and solutions converge to the original problem.

The proposed algorithm effectively solves the variational inequalities in numerical tests.

Numerical experiments confirm the efficiency and robustness of the method.

Abstract

This paper is concerned with the numerical solution of a class of variational inequalities of the second kind, involving the $p$-Laplacian operator. This kind of problems arise, for instance, in the mathematical modelling of non-Newtonian fluids. We study these problems by using a regularization approach, based on a Huber smoothing process. Well posedness of the regularized problems is proved, and convergence of the regularized solutions to the solution of the original problem is verified. We propose a preconditioned descent method for the numerical solution of these problems and analyze the convergence of this method in function spaces. The existence of admissible descent directions is established by variational methods and admissible steps are obtained by a backtracking algorithm which approximates the objective functional by polynomial models. Finally, several numerical experiments are carried out to show the efficiency of the methodology here introduced.