An optimal adaptive Fictitious Domain Method

TL;DR

This paper introduces an adaptive Fictitious Domain method for elliptic PDEs, combining saddle-point system approximation with adaptive finite element methods, achieving optimal convergence rates.

Contribution

It develops a new adaptive Fictitious Domain approach with an inexact preconditioned Uzawa algorithm, demonstrating optimal convergence theoretically and numerically.

Findings

Method converges with the best possible rate.

Numerical results confirm theoretical convergence.

Efficient approximation of elliptic problems using adaptive FEM.

Abstract

We consider a Fictitious Domain formulation of an elliptic partial differential equation and approximate the resulting saddle-point system using an inexact preconditioned Uzawa iterative algorithm. Each iteration entails the approximation of an elliptic problems performed using adaptive finite element methods. We prove that the overall method converges with the best possible rate and illustrate numerically our theoretical findings.