Analysis of the shifted boundary method for the Stokes problem

Atallah, Nabil M.; Canuto, Claudio; Scovazzi, Guglielmo

doi:10.1016/j.cma.2019.112609

Title: Analysis of the shifted boundary method for the Stokes problem

Abstract

The analysis of stability and accuracy of the shifted boundary method is developed for the Stokes flow equations. The key feature of the shifted boundary method, an embedded finite element method, is the shifting of the location where boundary conditions are applied from the true to a surrogate boundary, and an appropriate modification (shifting) of the value of the boundary conditions. An inf–sup condition is proved for the variational formulation associated to the shifted boundary method and we derive, by way of Strang’s second lemma, an optimal error estimate in the natural SBM norm. Here we also derive an L² -error estimate for the velocity field, by means of an extension of the Aubin–Nitsche approach to embedded, non-consistent, mixed finite element methods.

Authors:

Atallah, Nabil M. ^[1]; Canuto, Claudio ^[2];

^[1]

Duke University, Durham, NC (United States)
Polytechnic of Turin, Torino (Italy)

Publication Date:: Mon Sep 16 00:00:00 EDT 2019

Research Org.:: Duke Univ., Durham, NC (United States)

Sponsoring Org.:: USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR); Army Research Office (ARO)

OSTI Identifier:: 1802314

Alternate Identifier(s):: OSTI ID: 1562158

Grant/Contract Number:: SC0012169; W911NF-18-1-0308

Resource Type:: Accepted Manuscript

Journal Name:: Computer Methods in Applied Mechanics and Engineering

Additional Journal Information:: Journal Volume: 358; Journal Issue: C; Journal ID: ISSN 0045-7825

Publisher:: Elsevier

Country of Publication:: United States

Language:: English

Subject:: 97 MATHEMATICS AND COMPUTING; embedded methods; immersed boundary method; small cut-cell problem; approximate domain boundaries; weak boundary conditions; finite element method

Citation Formats


                    Atallah, Nabil M., Canuto, Claudio, and Scovazzi, Guglielmo. Analysis of the shifted boundary method for the Stokes problem.  United States: N. p., 2019. 
Web.  doi:10.1016/j.cma.2019.112609.

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                    Atallah, Nabil M., Canuto, Claudio, & Scovazzi, Guglielmo. Analysis of the shifted boundary method for the Stokes problem.  United States.  https://doi.org/10.1016/j.cma.2019.112609

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                    Atallah, Nabil M., Canuto, Claudio, and Scovazzi, Guglielmo. Mon .  
"Analysis of the shifted boundary method for the Stokes problem".  United States.  https://doi.org/10.1016/j.cma.2019.112609.  https://www.osti.gov/servlets/purl/1802314.

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@article{osti_1802314,

  title        = {Analysis of the shifted boundary method for the Stokes problem},

  author       = {Atallah, Nabil M. and Canuto, Claudio and Scovazzi, Guglielmo},

  abstractNote = {The analysis of stability and accuracy of the shifted boundary method is developed for the Stokes flow equations. The key feature of the shifted boundary method, an embedded finite element method, is the shifting of the location where boundary conditions are applied from the true to a surrogate boundary, and an appropriate modification (shifting) of the value of the boundary conditions. An inf–sup condition is proved for the variational formulation associated to the shifted boundary method and we derive, by way of Strang’s second lemma, an optimal error estimate in the natural SBM norm. Here we also derive an L2 -error estimate for the velocity field, by means of an extension of the Aubin–Nitsche approach to embedded, non-consistent, mixed finite element methods.},

  doi          = {10.1016/j.cma.2019.112609},

  journal      = {Computer Methods in Applied Mechanics and Engineering},

  number       = C,

  volume       = 358,

  place        = {United States},

  year         = {Mon Sep 16 00:00:00 EDT 2019},

  month        = {Mon Sep 16 00:00:00 EDT 2019}

}

Copy to clipboard

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Works referencing / citing this record:

Cut finite element error estimates for a class of nonlinear elliptic PDEs
preprint, January 2020

Katsouleas, Georgios; Karatzas, Efthymios N.; Travlopanos, Fotios
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The Second-Generation Shifted Boundary Method and Its Numerical Analysis
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Analysis of the Shifted Boundary Method for the Poisson Problem in General Domains
preprint, January 2020

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Title: Analysis of the shifted boundary method for the Stokes problem

Abstract

Citation Formats

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Stabilized mixed methods for the Stokes problem
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Solving Dirichlet Boundary-value Problems on Curved Domains by Extensions from Subdomains
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The shifted boundary method for embedded domain computations. Part II: Linear advection–diffusion and incompressible Navier–Stokes equations
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Shape optimization using the cut finite element method
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The shifted boundary method for hyperbolic systems: Embedded domain computations of linear waves and shallow water flows
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