A finite element method on composite grids based on Nitsche's method
ESAIM: Modélisation mathématique et analyse numérique, Tome 37 (2003) no. 3, pp. 495-514.

In this paper we propose a finite element method for the approximation of second order elliptic problems on composite grids. The method is based on continuous piecewise polynomial approximation on each grid and weak enforcement of the proper continuity at an artificial interface defined by edges (or faces) of one the grids. We prove optimal order a priori and energy type a posteriori error estimates in 2 and 3 space dimensions, and present some numerical examples.

DOI : 10.1051/m2an:2003039
Classification : 65N12, 65N30
Mots clés : Nitsche's method, overlapping grids
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     title = {A finite element method on composite grids based on {Nitsche's} method},
     journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
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Hansbo, Anita; Hansbo, Peter; Larson, Mats G. A finite element method on composite grids based on Nitsche's method. ESAIM: Modélisation mathématique et analyse numérique, Tome 37 (2003) no. 3, pp. 495-514. doi : 10.1051/m2an:2003039. http://www.numdam.org/articles/10.1051/m2an:2003039/

[1] Y. Achdou and Y. Maday, The mortar element method with overlapping subdomains. SIAM J. Numer. Anal. 40 (2002) 601-628. | Zbl

[2] R. Becker, P. Hansbo and R. Stenberg, A finite element method for domain decomposition with non-matching grids. ESAIM: M2AN 37 (2003) 209-225. | Numdam | Zbl

[3] M.J. Berger, On conservation at grid interfaces. SIAM J. Numer. Anal. 24 (1987) 967-984. | Zbl

[4] S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods. Springer-Verlag, Berlin (1994). | MR | Zbl

[5] F. Brezzi, J.-L. Lions and O. Pironneau, Analysis of a Chimera method. C. R. Acad. Sci. Paris Sér. I Math. 332 (2001) 655-660. | Zbl

[6] X.-C. Cai, M. Dryja and M. Sarkis, Overlapping nonmatching grid mortar element methods for elliptic problems. SIAM J. Numer. Anal. 36 (1999) 581-606. | Zbl

[7] G. Chesshire and W.D. Henshaw, Composite overlapping meshes for the solution of partial-differential equations. J. Comput. Phys. 90 (1990) 1-64. | Zbl

[8] V. Girault and P.-A. Raviart, Finite Element Approximation of the Navier-Stokes Equations. Springer-Verlag, Berlin (1979). | MR | Zbl

[9] A. Hansbo and P. Hansbo, An unfitted finite element method, based on Nitsche's method, for elliptic interface problems. Comput. Methods Appl. Mech. Engrg. 191 (2002) 5537-5552. | Zbl

[10] R.D. Lazarov, J.E. Pasciak, J. Schöberl and P.S. Vassilevski, Almost optimal interior penalty discontinuous approximations of symmetric elliptic problems on non-matching grids. Technical Report, ISC-01-05-MATH (2001). | Zbl

[11] R.D. Lazarov, S.Z. Tomov and P.S. Vassilevski, Interior penalty discontinuous approximations of elliptic problems. Comput. Meth. Appl. Math. 1 (2001) 367-382. | Zbl

[12] J. Nitsche, Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind. Abh. Math. Sem. Univ. Hamburg 36 (1971) 9-15. | Zbl

[13] L.R. Scott and S. Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comp. 190 (1990) 483-493. | Zbl

[14] R. Stenberg, Mortaring by a method of J.A. Nitsche, in Computational Mechanics: New Trends and Applications, S. Idelsohn, E. Onate and E. Dvorkin Eds., CIMNE, Barcelona (1998). | MR

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