Numerical Approximation of One Phase Quadrature Domains

TL;DR

This paper introduces two efficient numerical schemes for solving the free boundary problem of one phase quadrature domains, demonstrating their effectiveness through various numerical experiments.

Contribution

It presents two novel numerical methods, including an iterative solver and a shape reconstruction approach, for the first time applied to one phase quadrature domains.

Findings

The iterative solver is fast and adapts to topology changes.

Shape-Quasi-Newton-Method effectively reconstructs shapes.

Numerical experiments confirm the efficiency of both methods.

Abstract

In this work, we present two numerical schemes for a free boundary problem called one phase quadrature domain. In the first method by applying the proprieties of given free boundary problem, we derive a method that leads to a fast iterative solver. The iteration procedure is adapted in order to work in the case when topology changes. The second method is based on shape reconstruction to establish an efficient Shape-Quasi-Newton-Method. Various numerical experiments confirm the efficiency of the derived numerical methods.