Iterative solution of piecewise linear systems for the numerical solution of obstacle problems

TL;DR

This paper introduces semi-iterative Newton-type methods for solving piecewise linear systems that arise in obstacle problems, demonstrating their global convergence and effectiveness through numerical examples.

Contribution

The paper develops and proves the convergence of new semi-iterative Newton-type methods specifically designed for piecewise linear systems in obstacle problem applications.

Findings

Methods converge monotonically to the exact solution

Finite-step convergence demonstrated numerically

Effective for linear and parabolic obstacle problems

Abstract

We investigate the use of piecewise linear systems, whose coefficient matrix is a piecewise constant function of the solution itself. Such systems arise, for example, from the numerical solution of linear complementarity problems and in the numerical solution of free-surface problems. In particular, we here study their application to the numerical solution of both the (linear) parabolic obstacle problem and the obstacle problem. We propose a class of effective semi-iterative Newton-type methods to find the exact solution of such piecewise linear systems. We prove that the semiiterative Newton-type methods have a global monotonic convergence property, i.e., the iterates converge monotonically to the exact solution in a finite number of steps. Numerical examples are presented to demonstrate the effectiveness of the proposed methods.