Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-28T03:28:22.707Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  30 November 2010

Adrian J. Lew  and
Matteo Negri
Adrian J. Lew
Affiliation:
Department of Mechanical Engineering, Stanford University, Durand 207, Stanford 94305-4040, USA. [email protected]
Matteo Negri
Affiliation:
Department of Mathematics, University of Pavia, Via Ferrata 1, 27100 Pavia, Italy. [email protected]
Get access

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 C 2-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 h 2 and h, respectively. When q = 2, we have h 2-ε and h 1-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.

Keywords

Discontinuous Galerkinimmersed boundaryimmersed interface
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 45 , Issue 4 , July 2011 , pp. 651 - 674
Copyright
© EDP Sciences, SMAI, 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

R.A. Adams and J.J.F. Fournier, Sobolev spaces. Academic Press (2003).
Bramble, J.H. and King, J.T., A robust finite element method for nonhomogeneous Dirichlet problems in domains with curved boundaries. Math. Comp. 63 (1994) 1–17. CrossRef
Brezzi, F., Douglas, J. and Marini, L.D., Two families of mixed finite elements for second order elliptic problems. Numer. Math. 47 (1985) 217–235. CrossRef
Brezzi, F., Manzini, G., Marini, L.D., Pietra, P. and Russo, A., Discontinuous galerkin approximations for elliptic problems. Numer. Methods Partial Differ. Equ. 16 (2000) 365–378. 3.0.CO;2-Y>CrossRef
Brezzi, F., Hughes, T.J.R., Marini, L.D. and Masud, A., Mixed discontinuous Galerkin methods for Darcy flow. J. Sci. Comput. 22 (2005) 119–145. CrossRef
Burman, E. and Hansbo, P., Fictitious domain finite element methods using cut elements: I. A stabilized Lagrange multiplier method. Comput. Methods Appl. Mech. Eng. 199 (2010) 2680–2686. CrossRef
P.G. Ciarlet, The finite element method for elliptic problems. North-Holland (1978).
Codina, R. and Baiges, J., Approximate imposition of boundary conditions in immersed boundary methods. Int. J. Numer. Methods Eng. 80 (2009) 1379–1405. CrossRef
A. Ern and J.L. Guermond, Theory and practice of finite elements. Springer-Verlag (2004).
L.C. Evans and R.F. Gariepy, Measure theory and fine properties of functions. CRC (1992).
Girault, V. and Glowinski, R., Error analysis of a fictitious domain method applied to a Dirichlet problem. Japan J. Indust. Appl. Math. 12 (1995) 487–514. CrossRef
Glowinski, R., Pan, T.W. and Periaux, J., A fictitious domain method for Dirichlet problem and applications. Comput. Methods Appl. Mech. Eng. 111 (1994) 283–303. CrossRef
Hansbo, A. and Hansbo, P., An unfitted finite element method, based on Nitsche's method, for elliptic interface problems. Comput. Methods Appl. Mech. Eng. 191 (2002) 5537–5552. CrossRef
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).
Lenoir, M., Optimal isoparametric finite elements and error estimates for domains involving curved boundaries. SIAM J. Numer. Anal. 23 (1986) 562–580. CrossRef
Leveque, R.J. and The, Z. Li immersed interface method for elliptic equations with discontinuous coefficients and singular sources. SIAM J. Numer. Anal. 31 (1994) 1019–1044. CrossRef
Lew, A.J. and Buscaglia, G.C., A discontinuous-Galerkin-based immersed boundary method. Int. J. Numer. Methods Eng. 76 (2008) 427–454. CrossRef
Lew, A., Neff, P., Sulsky, D. and Ortiz, M., Optimal BV estimates for a discontinuous Galerkin method in linear elasticity. Appl. Math. Res. Express 3 (2004) 73–106. CrossRef
J.L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications. Springer-Verlag (1972).
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 Hamburg 36, Springer (1971) 9–15.
Rangarajan, R., Lew, A. and Buscaglia, G.C., 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. CrossRef
Thomee, V., Polygonal domain approximation in Dirichlet's problem. J. Inst. Math. Appl. 11 (1973) 33–44. CrossRef