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

top
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

top

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 -

References

top
  1. R.A. Adams and J.J.F. Fournier, Sobolev spaces. Academic Press (2003).  Zbl1098.46001
  2. J.H. Bramble and J.T. King, A robust finite element method for nonhomogeneous Dirichlet problems in domains with curved boundaries. Math. Comp.63 (1994) 1–17.  Zbl0810.65104
  3. F. Brezzi, J. Douglas and L.D. Marini, Two families of mixed finite elements for second order elliptic problems. Numer. Math.47 (1985) 217–235.  Zbl0599.65072
  4. F. Brezzi, G. Manzini, L.D. Marini, P. Pietra and A. Russo, Discontinuous galerkin approximations for elliptic problems. Numer. Methods Partial Differ. Equ.16 (2000) 365–378.  Zbl0957.65099
  5. F. Brezzi, T.J.R. Hughes, L.D. Marini and A. Masud, Mixed discontinuous Galerkin methods for Darcy flow. J. Sci. Comput.22 (2005) 119–145.  Zbl1103.76031
  6. E. Burman and P. Hansbo, Fictitious domain finite element methods using cut elements: I. A stabilized Lagrange multiplier method. Comput. Methods Appl. Mech. Eng.199 (2010) 2680–2686.  Zbl1231.65207
  7. P.G. Ciarlet, The finite element method for elliptic problems. North-Holland (1978).  Zbl0383.65058
  8. R. Codina and J. Baiges, Approximate imposition of boundary conditions in immersed boundary methods. Int. J. Numer. Methods Eng.80 (2009) 1379–1405.  Zbl1183.76802
  9. A. Ern and J.L. Guermond, Theory and practice of finite elements. Springer-Verlag (2004).  Zbl1059.65103
  10. L.C. Evans and R.F. Gariepy, Measure theory and fine properties of functions. CRC (1992).  Zbl0804.28001
  11. V. Girault and R. Glowinski, Error analysis of a fictitious domain method applied to a Dirichlet problem. Japan J. Indust. Appl. Math.12 (1995) 487–514.  Zbl0843.65076
  12. R. Glowinski, T.W. Pan and J. Periaux, A fictitious domain method for Dirichlet problem and applications. Comput. Methods Appl. Mech. Eng.111 (1994) 283–303.  Zbl0845.73078
  13. A. Hansbo and P. Hansbo, An unfitted finite element method, based on Nitsche's method, for elliptic interface problems. Comput. Methods Appl. Mech. Eng.191 (2002) 5537–5552.  Zbl1035.65125
  14. D. Henry, J. Hale and A.L. Pereira, Perturbation of the boundary in boundary-value problems of partial differential equations. Cambridge University Press, Cambridge (2005).  
  15. M. Lenoir, Optimal isoparametric finite elements and error estimates for domains involving curved boundaries. SIAM J. Numer. Anal.23 (1986) 562–580.  Zbl0605.65071
  16. R.J. Leveque and Z. Li, The immersed interface method for elliptic equations with discontinuous coefficients and singular sources. SIAM J. Numer. Anal.31 (1994) 1019–1044.  Zbl0811.65083
  17. A.J. Lew and G.C. Buscaglia, A discontinuous-Galerkin-based immersed boundary method. Int. J. Numer. Methods Eng.76 (2008) 427–454.  Zbl1195.76258
  18. A. Lew, P. Neff, D. Sulsky and M. Ortiz, Optimal BV estimates for a discontinuous Galerkin method in linear elasticity. Appl. Math. Res. Express3 (2004) 73–106.  Zbl1115.74021
  19. J.L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications. Springer-Verlag (1972).  
  20. J. Nitsche, Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind, in Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg36, Springer (1971) 9–15.  Zbl0229.65079
  21. R. Rangarajan, A. Lew and G.C. Buscaglia, A discontinuous-Galerkin-based immersed boundary method with non-homogeneous boundary conditions and its application to elasticity. Comput. Methods Appl. Mech. Eng.198 (2009) 1513–1534.  Zbl1227.74091
  22. V. Thomee, Polygonal domain approximation in Dirichlet's problem. J. Inst. Math. Appl.11 (1973) 33–44.  Zbl0246.35023

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.