Abstract
It is well-known that nonmatching grids are often used in finite element methods. Usually, grids are being matched along lines or surfaces that divide a domain into subdomains. Such lines or surfaces are called interfaces. The interface matching means the satisfaction of some continuity conditions when crossing the interface. The direct matching procedures fall into three groups: Methods that use Lagrange multipliers, mortar methods based on the Nitsche technique, and penalty methods.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Reference
I. Babuska, “Error-Bounds for Finite Element Method,” Numer. Math. 16, 322–333 (1971).
J. A. Nitsche, “Convergence of Nonconforming Methods,” Math. Aspects Finite Elem. Partial Differ. Equat., Proc. Symp. Madison, 15–53 (1974).
R. Becker, P. Hansbo, and R. Stenberg, “A Finite Element Method for Domain Decomposition with Non-Matching Grids,” Math. Model. Numer. Anal. 37(2), 209–225 (2003).
P. Le Tallec and T. Sassi, “Domain Decomposition with Nonmatching Grids: Augmented Lagrangian Approach,” Math. Comput. 64(212), 1367–1396 (1995).
I. Babuska, “The Finite Element Method with Penalty,” Math. Comput. 27, 221–228 (1973).
J.-P. Aubin, “Approximation des Problémes aux Limites non Homogénes et Régularité de la Convergence,” Calcolo 6, 117–140 (1969).
L. V. Maslovskaya and O. M. Maslovskaya, “Penalty Method for Grids Matching in Finite Element Method,” Izv. Vyssh. Uchebn. Zaved. Mat., No. 10, 33–43 (2006) [Russian Mathematics (Iz. VUZ) 50 (10), 29-39 (2006)].
L. V. Maslovskaya and O. M. Maslovskaya, “Some Methods of Grids Matching in Finite Elements Method,” in Materials of 7th All-Russia Workshop “Grid Methods for Boundary-Value Problems and Applications”, September 21-24, 2007, pp. 186–189.
L. V. Maslovskaya and O. M. Maslovskaya, “The Penalty Method for Grid Matching in Mixed Finite Element Methods,” Izv. Vyssh. Uchebn. Zaved. Mat., No. 3, 37–54 (2009). [Russian Mathematics (Iz. VUZ) 53 (3), 29-44 (2009)].
F. Brezzi and P. A. Raviart, “Mixed Finite Element Methods for 4th Order Elliptic Equations,” in Topics in Numerical Analysis III (Academic Press, London, New York, San Francisco, 1976), pp. 315–338.
R. S. Falk and J. E. Osborn, “Error Estimates for Mixed Methods,” RAIRO. Anal. Numér. 14(3), 249–277 (1980).
L. V. Maslovskaya, “Behavior of Solution of Boundary-Value Problems for Biharmonic Equations in Domains with Angular Points,” Differents. Uravneniya 19(12), 2172–2175 (1983).
Ph. Ciarlet, The Finite Element Method for Elliptic Problems (North Holland, Amsterdam, 1978; Mir, Moscow, 1980).
I. Babuška J. Osborn, and J. Pitkaeranta, “Analysis of Mixed Methods Using Mesh Dependent Norms,” Math. Comput. 35, 1039–1062 (1980).
F Brezzi, “On the Existence, Uniqueness and Approximation of Saddle-Point Problems Arising from Lagrangian Multipliers,” RAIRO, R28, 129–151 (1974).
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © L. V. Maslovskaya and O. M. Maslovskaya, 2010, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2010, No. 4, pp. 19–35.
About this article
Cite this article
Maslovskaya, L.V., Maslovskaya, O.M. The Nitsche mortar method for matching grids in a mixed finite element method. Russ Math. 54, 15–30 (2010). https://doi.org/10.3103/S1066369X10040031
Received:
Published:
Issue Date:
DOI: https://doi.org/10.3103/S1066369X10040031