The penalty method for grid matching in mixed finite element methods. (English. Russian original) Zbl 1178.65141

Russ. Math. 53, No. 3, 29-44 (2009); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 2009, No. 3, 37-54 (2009).
The finite element method of Hermann-Hellan-Johnson [F. Brezzi and P. A. Raviart, Mixed finite element methods for 4th order elliptic equations. Topics in numerical analysis III, Proc. R. Irish Acad. Conf., Dublin 1976, 33–56 (1977; Zbl 0434.65085); R. S. Falk and J. E. Osborn, RAIRO, Anal. Numér. 14, 249–277 (1980; Zbl 0467.65062)] for the biharmonic equation is considered for non-matching grids. Matching conditions are replaced by penalty terms. The convergence rate is less than that of the mixed finite element method on matching grids with any choice of penalty parameters. This is in contrast to results by the authors for other finite element methods.


65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
Full Text: DOI


[1] I. Babuska, ”Error-Bounds for Finite Element Method,” Numer.Math. 16, 322–333 (1971). · Zbl 0214.42001
[2] J. A. Nitsche, ”Convergence of Nonconforming Methods,” in Math. Aspects of Finite Elements in Partial Differential Equations (Academic Press, New York, 1974), pp. 15–53.
[3] R. Becker, P. Hansbo, and R. Stenberg, ”A Finite Element Method for Domain Decomposition with Non-Matching Grids,” Math. Model. Numer. Anal. 37(212), 209–225 (2003). · Zbl 1047.65099
[4] P. Le Tallec and T. Sassi, ”Domain Decomposition with Nonmatching Grids: Augmented Lagrangian Approach,” Math. Comp. 64(212), 1367–1396 (1995). · Zbl 0849.65087
[5] I. Babuska, ”The Finite Element Method with Penalty,” Math. Comp. 27(122), 221–228 (1973).
[6] J. P. Aubin, ”Approximation des Problems aux Limites non Homogenes et Regularite de la Convergence,” Calcolo 6, 117–139 (1969). · Zbl 0201.12601
[7] L. V. Maslovskaya and O. M. Maslovskaya, ”Penalty Method for Grids Matching in the Finite Element Method,” Izv. Vyssh. Uchebn. Zaved. Mat., No. 10, 33–43 (2006) [Russian Mathematics (Iz. VUZ) 50 (10), 29–39 (2006)]. · Zbl 0663.73043
[8] F. Brezzi and P. A. Raviart, ”Mixed Finite Element Methods for 4th Order Elliptic Equations,” in Topics in Numerical Analysis (Academic Press, New York, 1976), No. 3, pp. 315–338. · Zbl 0434.65085
[9] R. S. Falk and J. E. Osborn, ”Error Estimates for Mixed Methods,” R. A. I. R. O. Anal. Numer. 14(3), 249–277 (1980). · Zbl 0467.65062
[10] L. V. Maslovskaya, ”Behavior of Solution of Boundary Value Problems for Biharmonic Equations in Domains with Angular Points,” Differents. Uravn. 19(2), 2172–2175 (1983). · Zbl 0551.35026
[11] Ph. Ciarlet The Finite Element Method for Elliptic Problems, (North-Holland Publishing Co., Amsterdam, 1978; Mir, Moscow, 1980).
[12] I. Babuska, J. Osborn, and J. Pitkaranta, ”Analysis of Mixed Methods Using Mesh Dependent Norms,” Math. Comp. 35, 1039–1062 (1980).
[13] F. Brezzi, ”On the Existence, Uniqueness and Approximation of Saddle-Point Problems Arising from Lagrangian Multipliers,” R. A. I. R. O., R2. 8, 129–151 (1974). · Zbl 0338.90047
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.