Hostname: page-component-788cddb947-2s2w2 Total loading time: 0 Render date: 2024-10-14T22:23:43.371Z Has data issue: false hasContentIssue false

A finite element method on composite grids based on Nitsche's method

Published online by Cambridge University Press:  15 April 2004

Anita Hansbo
Affiliation:
Department of Informatics and Mathematics, University of Trollhättan Uddevalla, Box 957, 461 39 Trollhättan, Sweden. hansbo@solid.chalmers.se.
Peter Hansbo
Affiliation:
Department of Applied Mechanics, Chalmers University of Technology, 412 96 Göteborg, Sweden.
Mats G. Larson
Affiliation:
Department of Mathematics, Chalmers University of Technology, 412 96 Göteborg, Sweden.
Get access

Abstract

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.

Type
Research Article
Copyright
© EDP Sciences, SMAI, 2003

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

Achdou, Y. and Maday, Y., The mortar element method with overlapping subdomains. SIAM J. Numer. Anal. 40 (2002) 601628. CrossRef
Becker, R., Hansbo, P. and Stenberg, R., A finite element method for domain decomposition with non-matching grids. ESAIM: M2AN 37 (2003) 209225. CrossRef
Berger, M.J., On conservation at grid interfaces. SIAM J. Numer. Anal. 24 (1987) 967984. CrossRef
S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods. Springer-Verlag, Berlin (1994).
Brezzi, F., Lions, J.-L. and Pironneau, O., Analysis of a Chimera method. C. R. Acad. Sci. Paris Sér. I Math. 332 (2001) 655660. CrossRef
Cai, X.-C., Dryja, M. and Sarkis, M., Overlapping nonmatching grid mortar element methods for elliptic problems. SIAM J. Numer. Anal. 36 (1999) 581606. CrossRef
Chesshire, G. and Henshaw, W.D., Composite overlapping meshes for the solution of partial-differential equations. J. Comput. Phys. 90 (1990) 164. CrossRef
V. Girault and P.-A. Raviart, Finite Element Approximation of the Navier-Stokes Equations. Springer-Verlag, Berlin (1979).
Hansbo, A. and Hansbo, P., An unfitted finite element method, based on Nitsche's method, for elliptic interface problems. Comput. Methods Appl. Mech. Engrg. 191 (2002) 55375552. CrossRef
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).
Lazarov, R.D., Tomov, S.Z. and Vassilevski, P.S., Interior penalty discontinuous approximations of elliptic problems. Comput. Meth. Appl. Math. 1 (2001) 367382. CrossRef
Nitsche, J., Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind. Abh. Math. Sem. Univ. Hamburg 36 (1971) 915. CrossRef
Scott, L.R. and Zhang, S., Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comp. 190 (1990) 483493. CrossRef
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).