A Relaxed Ka\v{c}anov Iteration for the $p$-Poisson Problem

TL;DR

This paper introduces a relaxed Kacanov iteration method for approximating solutions to the $p$-Poisson problem with $1 < p \,\leq \, 2$, proving algebraic convergence rates.

Contribution

It presents a new iterative algorithm that simplifies solving the $p$-Poisson problem and provides convergence analysis for the method.

Findings

Algorithm converges with an algebraic rate.

Method effectively approximates weak solutions.

Applicable for $p$-Poisson problems with $1 < p \,\leq \, 2$.

Abstract

In this paper we introduce and analyze an iteratively re-weighted algorithm, that allows to approximate the weak solution of the $p$-Poisson problem for $1 < p \leq 2$ by iteratively solving a sequence of linear elliptic problems. The algorithm can be interpreted as a relaxed Ka{\v c}anov iteration, as so-called in the specific literature of the numerical solution of quasi-linear equations. The main contribution of the paper is proving that the algorithm converges at least with an algebraic rate.