Higher-order finite element methods for elliptic problems with interfaces
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 50 (2016) no. 5, pp. 1561-1583.

We present higher-order piecewise continuous finite element methods for solving a class of interface problems in two dimensions. The method is based on correction terms added to the right-hand side in the standard variational formulation of the problem. We prove optimal error estimates of the methods on general quasi-uniform and shape regular meshes in maximum norms. In addition, we apply the method to a Stokes interface problem, adding correction terms for the velocity and the pressure, obtaining optimal convergence results.

Received:
Accepted:
DOI: 10.1051/m2an/2015093
Classification: 65N30, 65N15
Keywords: Interface problems, finite elements, pointwise estimates
Guzmán, Johnny 1; Sánchez, Manuel A. 1; Sarkis, Marcus 2

1 Division of Applied Mathematics, Brown University, Providence, RI 02912, USA
2 Department of Mathematical Sciences at Worcester Polytechnic Institute, 100 Institute Road, Worcester, MA 01609, USA
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Guzmán, Johnny; Sánchez, Manuel A.; Sarkis, Marcus. Higher-order finite element methods for elliptic problems with interfaces. ESAIM: Mathematical Modelling and Numerical Analysis , Volume 50 (2016) no. 5, pp. 1561-1583. doi : 10.1051/m2an/2015093. http://www.numdam.org/articles/10.1051/m2an/2015093/

S. Adjerid, M. Ben-Romdhane and T. Lin, Higher degree immersed finite element methods for second-order elliptic interface problems. Int. J. Numer. Anal. Model. 11 (2014) 541–566. | MR | Zbl

C. Annavarapu, M. Hautefeuille and J.E. Dolbow, A robust Nitsche’s formulation for interface problems. Comput. Methods Appl. Mech. Engrg. 225/228 (2012) 44–54. | DOI | MR | Zbl

J.T. Beale and A.T. Layton, On the accuracy of finite difference methods for elliptic problems with interfaces. Commun. Appl. Math. Comput. Sci. 1 (2006) 91–119. | DOI | MR | Zbl

J. Bedrossian, J.H. Von Brecht, S. Zhu, E. Sifakis and J.M. Teran, A second order virtual node method for elliptic problems with interfaces and irregular domains. J. Comput. Phys. 229 (2010) 6405–6426. | DOI | MR | Zbl

D. Boffi and L. Gastaldi, A finite element approach for the immersed boundary method. Comput. Struct. 81 (2003) 491–501. In honour of Klaus-Jürgen Bathe. | DOI | MR

D. Boffi, N. Cavallini and L. Gastaldi, Finite element approach to immersed boundary method with different fluid and solid densities. Math. Models Methods Appl. Sci. 21 (2011) 2523–2550. | DOI | MR | Zbl

S.C. Brenner and L.R. Scott, The mathematical theory of finite element methods. Vol. 15 of Texts in Applied Mathematics. Springer-Verlag, New York (1994). | MR | Zbl

E. Burman, Ghost penalty. C. R. Math. Acad. Sci. Paris 348 (2010) 1217–1220. | DOI | MR | Zbl

E. Burman, Projection stabilization of Lagrange multipliers for the imposition of constraints on interfaces and boundaries. Numer. Methods Partial Differ. Equ. 30 (2014) 567–592. | DOI | MR | Zbl

E. Burman and P. Hansbo, Fictitious domain finite element methods using cut elements: I. A stabilized Lagrange multiplier method. Comput. Methods Appl. Mech. Engrg. 199 (2010) 2680–2686. | DOI | MR | Zbl

E. Burman and P. Hansbo, Fictitious domain finite element methods using cut elements: II. A stabilized Nitsche method. Appl. Numer. Math. 62 (2012) 328–341. | DOI | MR | Zbl

E. Burman and P. Zunino, Numerical approximation of large contrast problems with the unfitted Nitsche method. In Frontiers in numerical analysis – Durham 2010. Vol. 85 of Lect. Notes Comput. Sci. Eng. Springer, Heidelberg (2012) 227–282. | MR | Zbl

C.-C. Chu, I.G. Graham and T.-Y. Hou, A new multiscale finite element method for high-contrast elliptic interface problems. Math. Comput. 79 (2010) 1915–1955. | DOI | MR | Zbl

R. Cortez, L. Fauci, N. Cowen and R. Dillon, Simulation of swimming organisms: Coupling internal mechanics with external fluid dynamics. Comput. Sci. Eng. 6 (2004) 38–45. | DOI

A. Demlow, D. Leykekhman, A.H. Schatz and L.B. Wahlbin, Best approximation property in the W 1 norm for finite element methods on graded meshes. Math. Comput. 81 (2012) 743–764. | DOI | MR | Zbl

D. Gilbarg and N.S. Trudinger, Elliptic partial differential equations of second order. Classics in Mathematics. Reprint of the 1998 edition. Springer-Verlag, Berlin (2001). | MR | Zbl

V. Girault, R.H. Nochetto and R. Scott, Stability of the finite element Stokes projection in W 1, . C. R. Math. Acad. Sci. Paris 338 (2004) 957–962. | DOI | MR | Zbl

Y. Gong, B. Li and Z. Li, Immersed-interface finite-element methods for elliptic interface problems with nonhomogeneous jump conditions. SIAM J. Numer. Anal. 46 (2007/08) 472–495. | DOI | MR | Zbl

J. Guzmán and D. Leykekhman, Pointwise error estimates of finite element approximations to the Stokes problem on convex polyhedra. Math. Comput. 81 (2012) 1879–1902. | DOI | MR | Zbl

X. He, T. Lin and Y. Lin, Immersed finite element methods for elliptic interface problems with non-homogeneous jump conditions. Int. J. Numer. Anal. Model. 8 (2011) 284–301. | MR | Zbl

S. Hou and X.-D. Liu, A numerical method for solving variable coefficient elliptic equation with interfaces. J. Comput. Phys. 202 (2005) 411–445. | DOI | MR | Zbl

S. Hou, W. Wang and L. Wang, Numerical method for solving matrix coefficient elliptic equation with sharp-edged interfaces. J. Comput. Phys. 229 (2010) 7162–7179. | DOI | MR | Zbl

S. Hou, P. Song, L. Wang and H. Zhao, A weak formulation for solving elliptic interface problems without body fitted grid. J. Comput. Phys. 249 (2013) 80–95. | DOI | MR | Zbl

M. Sanchez-Uribe, J. Guzman and M. Sarkis, On the accuracy of finite element approximations to a class of interface problems. Technical Report 2014-6, Scientific Computing Group, Brown University, Providence, RI, USA, March (2014).

L. Lee and R.J. Leveque, An immersed interface method for incompressible Navier-Stokes equations. SIAM J. Sci. Comput. 25 (2003) 832–856. | DOI | MR | Zbl

R.J. Leveque and Z.L. Li, The immersed interface method for elliptic equations with discontinuous coefficients and singular sources. SIAM J. Numer. Anal. 31 (1994) 1019–1044. | DOI | MR | Zbl

R.J. Leveque and Z. Li, Immersed interface methods for Stokes flow with elastic boundaries or surface tension. SIAM J. Sci. Comput. 18 (1997) 709–735. | DOI | MR | Zbl

Z. Li and M.-C. Lai, The immersed interface method for the Navier–Stokes equations with singular forces. J. Comput. Phys. 171 (2001) 822–842. | DOI | MR | Zbl

A.N. Marques, J.-C. Nave and R.R. Rosales, A correction function method for Poisson problems with interface jump conditions. J. Comput. Phys. 230 (2011) 7567–7597. | DOI | MR | Zbl

C.S. Peskin, Numerical analysis of blood flow in the heart. J. Comput. Phys. 25 (1977) 220–252. | DOI | MR | Zbl

C.S. Peskin, The immersed boundary method. Acta Numer. 11 (2002) 479–517. | DOI | MR | Zbl

R. Rannacher and R. Scott, Some optimal error estimates for piecewise linear finite element approximations. Math. Comput. 38 (1982) 437–445. | DOI | MR | Zbl

P.-A. Raviart and J.M. Thomas, A mixed finite element method for 2nd order elliptic problems. In Mathematical aspects of finite element methods (Proc. Conf., Consiglio Naz. delle Ricerche, C.N.R., Rome, 1975). Vol. 606 of Lect. Notes Math. Springer, Berlin (1977) 292–315. | MR | Zbl

C. Tu and C.S. Peskin, Stability and instability in the computation of flows with moving immersed boundaries: a comparison of three methods. SIAM J. Sci. Statist. Comput. 13 (1992) 1361–1376. | DOI | MR | Zbl

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