Analysis of the shifted boundary method for the Poisson problem in domains with corners
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- by Nabil M. Atallah, Claudio Canuto and Guglielmo Scovazzi HTML | PDF
- Math. Comp. 90 (2021), 2041-2069 Request permission
Abstract:
The shifted boundary method (SBM) is an approximate domain method for boundary value problems, in the broader class of unfitted/embedded/immersed methods. It has proven to be quite efficient in handling problems with complex geometries, ranging from Poisson to Darcy, from Navier-Stokes to elasticity and beyond. The key feature of the SBM is a shift in the location where Dirichlet boundary conditions are applied—from the true to a surrogate boundary—and an appropriate modification (again, a shift) of the value of the boundary conditions, in order to reduce the consistency error. In this paper we provide a sound analysis of the method in smooth domains and in domains with corners, highlighting the influence of geometry and distance between exact and surrogate boundaries upon the convergence rate. We consider the Poisson problem with Dirichlet boundary conditions as a model and we first detail a procedure to obtain the crucial shifting between the surrogate and the true boundaries. Next, we give a sufficient condition for the well-posedness and stability of the discrete problem. The behavior of the consistency error arising from shifting the boundary conditions is thoroughly analyzed, for smooth boundaries and for boundaries with corners and edges. The convergence rate is proven to be optimal in the energy norm, and is further enhanced in the $L^2$-norm.References
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Additional Information
- Nabil M. Atallah
- Affiliation: Department of Civil and Environmental Engineering, Duke University, Durham, North Carolina 27708
- MR Author ID: 1343085
- Email: nabil.atallah@duke.edu
- Claudio Canuto
- Affiliation: Dipartimento di Scienze Matematiche, Politecnico di Torino, Corso Duca degli Abruzzi 24, I-10129 Torino, Italy
- MR Author ID: 44965
- ORCID: 0000-0002-8481-0312
- Email: claudio.canuto@polito.it
- Guglielmo Scovazzi
- Affiliation: Department of Civil and Environmental Engineering, Duke University, Durham, North Carolina 27708
- MR Author ID: 780205
- ORCID: 0000-0003-1043-6281
- Email: guglielmo.scovazzi@duke.edu
- Received by editor(s): May 6, 2020
- Received by editor(s) in revised form: December 3, 2020
- Published electronically: May 6, 2021
- Additional Notes: The support of the Army Research Office (ARO) under Grant W911NF-18-1-0308 was gratefully acknowledged. The second author performed this research in the framework of the Italian MIUR Award “Dipartimenti di Eccellenza 2018-2022” granted to the Department of Mathematical Sciences, Politecnico di Torino (CUP: E11G18000350001), and with the support of the Italian MIUR PRIN Project 201752HKH8-003. He is a member of the Italian INdAM-GNCS research group.
- © Copyright 2021 American Mathematical Society
- Journal: Math. Comp. 90 (2021), 2041-2069
- MSC (2020): Primary 65N30; Secondary 65N12, 65N50
- DOI: https://doi.org/10.1090/mcom/3641
- MathSciNet review: 4280292