An optimal adaptive Fictitious Domain Method

Berrone, Stefano; Bonito, Andrea; Stevenson, Rob; Verani, Marco

doi:10.1090/mcom/3414

An optimal adaptive Fictitious Domain Method
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by Stefano Berrone, Andrea Bonito, Rob Stevenson and Marco Verani;

Math. Comp. 88 (2019), 2101-2134

DOI: https://doi.org/10.1090/mcom/3414

Published electronically: February 22, 2019

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Abstract:

We consider a fictitious domain formulation of an elliptic partial differential equation and approximate the resulting saddle-point system using a nested inexact preconditioned Uzawa iterative algorithm, which consists of three nested loops. In the outer loop the trial space for the Galerkin approximation of the Lagrange multiplier is enlarged. The intermediate loop solves this Galerkin system by a damped preconditioned Richardson iteration. Each iteration of the latter involves solving an elliptic problem on the fictitious domain whose solution is approximated by an adaptive finite element method in the inner loop. We prove that the overall method converges with the best possible rate and illustrate numerically our theoretical findings.

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