Optimal convergence of a discontinuous-Galerkin-based immersed boundary method
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 45 (2011) no. 4, p. 651-674

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.

DOI : https://doi.org/10.1051/m2an/2010069
Classification:  65N30,  65N15
Keywords: discontinuous Galerkin, immersed boundary, immersed interface
@article{M2AN_2011__45_4_651_0,
     author = {Lew, Adrian J. and Negri, Matteo},
     title = {Optimal convergence of a discontinuous-Galerkin-based immersed boundary method},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {45},
     number = {4},
     year = {2011},
     pages = {651-674},
     doi = {10.1051/m2an/2010069},
     zbl = {1269.65108},
     mrnumber = {2804654},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2011__45_4_651_0}
}
Lew, Adrian J.; Negri, Matteo. Optimal convergence of a discontinuous-Galerkin-based immersed boundary method. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 45 (2011) no. 4, pp. 651-674. doi : 10.1051/m2an/2010069. http://www.numdam.org/item/M2AN_2011__45_4_651_0/

[1] R.A. Adams and J.J.F. Fournier, Sobolev spaces. Academic Press (2003). | MR 2424078 | Zbl 1098.46001

[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. | MR 1242055 | Zbl 0810.65104

[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. | MR 799685 | Zbl 0599.65072

[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. | MR 1765651 | Zbl 0957.65099

[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. | MR 2142192 | Zbl 1103.76031

[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. | MR 2728820 | Zbl 1231.65207

[7] P.G. Ciarlet, The finite element method for elliptic problems. North-Holland (1978). | MR 520174 | Zbl 0511.65078

[8] R. Codina and J. Baiges, Approximate imposition of boundary conditions in immersed boundary methods. Int. J. Numer. Methods Eng. 80 (2009) 1379-1405. | MR 2582494 | Zbl 1183.76802

[9] A. Ern and J.L. Guermond, Theory and practice of finite elements. Springer-Verlag (2004). | MR 2050138 | Zbl 1059.65103

[10] L.C. Evans and R.F. Gariepy, Measure theory and fine properties of functions. CRC (1992). | MR 1158660 | Zbl 0804.28001

[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. | MR 1356667 | Zbl 0843.65076

[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. | MR 1259864 | Zbl 0845.73078

[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. | MR 1941489 | Zbl 1035.65125

[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). | MR 2160744 | Zbl 1170.35300

[15] M. Lenoir, Optimal isoparametric finite elements and error estimates for domains involving curved boundaries. SIAM J. Numer. Anal. 23 (1986) 562-580. | MR 842644 | Zbl 0605.65071

[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. | MR 1286215 | Zbl 0811.65083

[17] A.J. Lew and G.C. Buscaglia, A discontinuous-Galerkin-based immersed boundary method. Int. J. Numer. Methods Eng. 76 (2008) 427-454. | MR 2462713 | Zbl 1195.76258

[18] A. Lew, P. Neff, D. Sulsky and M. Ortiz, Optimal BV estimates for a discontinuous Galerkin method in linear elasticity. Appl. Math. Res. Express 3 (2004) 73-106. | MR 2091832 | Zbl 1115.74021

[19] J.L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications. Springer-Verlag (1972). | Zbl 0223.35039

[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 Hamburg 36, Springer (1971) 9-15. | MR 341903 | Zbl 0229.65079

[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. | MR 2513367 | Zbl 1227.74091

[22] V. Thomee, Polygonal domain approximation in Dirichlet's problem. J. Inst. Math. Appl. 11 (1973) 33-44. | MR 349044 | Zbl 0246.35023