Optimal convergence of a discontinuous-Galerkin-based immersed boundary method*

Adrian J. Lew; Matteo Negri

ESAIM: Mathematical Modelling and Numerical Analysis (2011)

  • Volume: 45, Issue: 4, page 651-674
  • ISSN: 0764-583X

Abstract

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We prove the optimal convergence of a discontinuous-Galerkin-based immersed boundary method introduced earlier [Lew and Buscaglia, Int. J. Numer. Methods Eng.76 (2008) 427–454]. By switching to a discontinuous Galerkin discretization near the boundary, this method overcomes the suboptimal convergence rate that may arise in immersed boundary methods when strongly imposing essential boundary conditions. We consider a model Poisson's problem with homogeneous boundary conditions over two-dimensional C2-domains. For solution in Hq for q > 2, we prove that the method constructed with polynomials of degree one on each element approximates the function and its gradient with optimal orders h2 and h, respectively. When q = 2, we have h2-ε and h1-ε for any ϵ > 0 instead. To this end, we construct a new interpolant that takes advantage of the discontinuities in the space, since standard interpolation estimates lead here to suboptimal approximation rates. The interpolation error estimate is based on proving an analog to Deny-Lions' lemma for discontinuous interpolants on a patch formed by the reference elements of any element and its three face-sharing neighbors. Consistency errors arising due to differences between the exact and the approximate domains are treated using Hardy's inequality together with more standard results on Sobolev functions.

How to cite

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Lew, Adrian J., and Negri, Matteo. "Optimal convergence of a discontinuous-Galerkin-based immersed boundary method*." ESAIM: Mathematical Modelling and Numerical Analysis 45.4 (2011): 651-674. <http://eudml.org/doc/276345>.

@article{Lew2011,
abstract = { We prove the optimal convergence of a discontinuous-Galerkin-based immersed boundary method introduced earlier [Lew and Buscaglia, Int. J. Numer. Methods Eng.76 (2008) 427–454]. By switching to a discontinuous Galerkin discretization near the boundary, this method overcomes the suboptimal convergence rate that may arise in immersed boundary methods when strongly imposing essential boundary conditions. We consider a model Poisson's problem with homogeneous boundary conditions over two-dimensional C2-domains. For solution in Hq for q > 2, we prove that the method constructed with polynomials of degree one on each element approximates the function and its gradient with optimal orders h2 and h, respectively. When q = 2, we have h2-ε and h1-ε for any ϵ > 0 instead. To this end, we construct a new interpolant that takes advantage of the discontinuities in the space, since standard interpolation estimates lead here to suboptimal approximation rates. The interpolation error estimate is based on proving an analog to Deny-Lions' lemma for discontinuous interpolants on a patch formed by the reference elements of any element and its three face-sharing neighbors. Consistency errors arising due to differences between the exact and the approximate domains are treated using Hardy's inequality together with more standard results on Sobolev functions. },
author = {Lew, Adrian J., Negri, Matteo},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Discontinuous Galerkin; immersed boundary; immersed interface; discontinuous Galerkin; error bounds; convergence; Poisson's problem; discontinuous interpolants; consistency},
language = {eng},
month = {1},
number = {4},
pages = {651-674},
publisher = {EDP Sciences},
title = {Optimal convergence of a discontinuous-Galerkin-based immersed boundary method*},
url = {http://eudml.org/doc/276345},
volume = {45},
year = {2011},
}

TY - JOUR
AU - Lew, Adrian J.
AU - Negri, Matteo
TI - Optimal convergence of a discontinuous-Galerkin-based immersed boundary method*
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2011/1//
PB - EDP Sciences
VL - 45
IS - 4
SP - 651
EP - 674
AB - We prove the optimal convergence of a discontinuous-Galerkin-based immersed boundary method introduced earlier [Lew and Buscaglia, Int. J. Numer. Methods Eng.76 (2008) 427–454]. By switching to a discontinuous Galerkin discretization near the boundary, this method overcomes the suboptimal convergence rate that may arise in immersed boundary methods when strongly imposing essential boundary conditions. We consider a model Poisson's problem with homogeneous boundary conditions over two-dimensional C2-domains. For solution in Hq for q > 2, we prove that the method constructed with polynomials of degree one on each element approximates the function and its gradient with optimal orders h2 and h, respectively. When q = 2, we have h2-ε and h1-ε for any ϵ > 0 instead. To this end, we construct a new interpolant that takes advantage of the discontinuities in the space, since standard interpolation estimates lead here to suboptimal approximation rates. The interpolation error estimate is based on proving an analog to Deny-Lions' lemma for discontinuous interpolants on a patch formed by the reference elements of any element and its three face-sharing neighbors. Consistency errors arising due to differences between the exact and the approximate domains are treated using Hardy's inequality together with more standard results on Sobolev functions.
LA - eng
KW - Discontinuous Galerkin; immersed boundary; immersed interface; discontinuous Galerkin; error bounds; convergence; Poisson's problem; discontinuous interpolants; consistency
UR - http://eudml.org/doc/276345
ER -

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