Abstract
In this paper we prove the possibility of the use of the penalty method for grid matching in mixed finite element methods. We consider the Hermann-Johnson scheme for biharmonic equation. The main idea is to construct a perturbed problem with two parameters which play roles of penalties. The perturbed problem is built by the replacement of essential conditions on the interface in the mixed variational statement with natural conditions that contain parameters. The perturbed problem is discretized by the finite element method. We estimate the norm of the difference between a solution of the discrete perturbed problem and a solution of the initial problem; the obtained estimates depend on the step and the penalties. We give recommendations for the choice of penalties depending on the step.
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References
I. Babuska, “Error-Bounds for Finite Element Method,” Numer.Math. 16, 322–333 (1971).
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.
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).
P. Le Tallec and T. Sassi, “Domain Decomposition with Nonmatching Grids: Augmented Lagrangian Approach,” Math. Comp. 64(212), 1367–1396 (1995).
I. Babuska, “The Finite Element Method with Penalty,” Math. Comp. 27(122), 221–228 (1973).
J. P. Aubin, “Approximation des Problems aux Limites non Homogenes et Regularite de la Convergence,” Calcolo 6, 117–139 (1969).
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)].
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.
R. S. Falk and J. E. Osborn, “Error Estimates for Mixed Methods,” R. A. I. R. O. Anal. Numer. 14(3), 249–277 (1980).
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).
Ph. Ciarlet The Finite Element Method for Elliptic Problems, (North-Holland Publishing Co., Amsterdam, 1978; Mir, Moscow, 1980).
I. Babuska, J. Osborn, and J. Pitkaranta, “Analysis of Mixed Methods Using Mesh Dependent Norms,” Math. Comp. 35, 1039–1062 (1980).
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).
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Original Russian Text © L.V. Maslovskaya, O.M. Maslovskaya, 2009, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2009, No. 3, pp. 37–54.
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Maslovskaya, L.V., Maslovskaya, O.M. The penalty method for grid matching in mixed finite element methods. Russ Math. 53, 29–44 (2009). https://doi.org/10.3103/S1066369X09030025
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DOI: https://doi.org/10.3103/S1066369X09030025