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Mathematics of Computation

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An optimal adaptive Fictitious Domain Method


Authors: Stefano Berrone, Andrea Bonito, Rob Stevenson and Marco Verani
Journal: Math. Comp. 88 (2019), 2101-2134
MSC (2010): Primary 41A25, 42C40, 65N12, 65T60, 65M85, 65N30
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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Additional Information

Stefano Berrone
Affiliation: Dipartimento di Scienze Matematiche, Politecnico di Torino, Corso Duca degli Abruzzi, 24, 10129 Torino, Italy
Email: stefano.berrone@polito.it

Andrea Bonito
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843
Email: bonito@math.tamu.edu

Rob Stevenson
Affiliation: Korteweg-de Vries Institute for Mathematics, University of Amsterdam, P.O. Box 94248, 1090 GE Amsterdam, The Netherlands
Email: r.p.stevenson@uva.nl

Marco Verani
Affiliation: MOX-Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo Da Vinci, 32, 20133 Milano, Italy
Email: marco.verani@polimi.it

DOI: https://doi.org/10.1090/mcom/3414
Keywords: Fictitious Domain Method, Uzawa iteration, adaptive finite element method, optimal preconditioner, best possible convergence rate
Received by editor(s): December 26, 2017
Received by editor(s) in revised form: September 2, 2018, and November 2, 2018
Published electronically: February 22, 2019
Additional Notes: The first author was partially supported by INdAM-GNCS and HPC@polito.it
The second author was partially supported by NSF Grant DMS-1254618.
The fourth author was partially supported by INdAM-GNCS
Article copyright: © Copyright 2019 American Mathematical Society