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

ESAIM: Mathematical Modelling and Numerical Analysis (2011)

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

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topLew, 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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