Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Local stability conditions for the Babuška method of Lagrange multipliers

Author: Juhani Pitkäranta
Journal: Math. Comp. 35 (1980), 1113-1129
MSC: Primary 65N30
MathSciNet review: 583490
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We consider the so-called Babuška method of finite elements with Lagrange multipliers for numerically solving the problem $ \Delta u = f$ in $ \Omega $, $ u = g$ on $ \partial \Omega $, $ \Omega \subset {R^n}$, $ n \geqslant 2$. We state a number of local conditions from which we prove the uniform stability of the Lagrange multiplier method in terms of a weighted, mesh-dependent norm. The stability conditions given weaken the conditions known so far and allow mesh refinements on the boundary. As an application, we introduce a class of finite element schemes, for which the stability conditions are satisfied, and we show that the convergence rate of these schemes is of optimal order.

References [Enhancements On Off] (What's this?)

  • [1] I. BABUSKA, "The finite element method with Lagrange multipliers," Numer. Math., v. 20, 1973, pp. 179-192. MR 0359352 (50:11806)
  • [2] I. BABUŠKA & A. K. AZIZ, "Survey lectures on the mathematical foundations of the finite element method," in The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations (A. K. Aziz, Ed.), Academic Press, New York, 1972. MR 0347104 (49:11824)
  • [3] I. BABUŠKA, R. B. KELLOGG & J. PITKÄRANTA, "Direct and inverse error estimates for finite elements with mesh refinements," Numer. Math., v. 33, 1979, pp. 447-471. MR 553353 (81c:65054)
  • [4] F. BREZZI, "On the existence, uniqueness and approximations of saddle-point problems arising from Lagrange multipliers," RAIRO, Sèr. Anal. Numér., v. 8, 1974, pp. 129-151. MR 0365287 (51:1540)
  • [5] P. G. CIARLET, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978. MR 0520174 (58:25001)
  • [6] J. DOUGLAS, JR., T. DUPONT & L. WAHLBIN, "The stability in $ {L^q}$ of the $ {L_2}$-projection into finite element function spaces," Numer. Math., v. 23, 1975, pp. 193-197. MR 0383789 (52:4669)
  • [7] J. L. LIONS & E. MAGENES, Problèmes aux Limites Non Homogènes et Applications, Vol. 1, Dunod, Paris, 1968. MR 0247243 (40:512)
  • [8] J. PITKÄRANTA, "Boundary subspaces for the finite element method with Lagrange multipliers," Numer. Math., v. 33, 1979, pp. 273-289. MR 553590 (82b:65138)
  • [9] J. PITKÄRANTA, The Finite Element Method with Lagrange Multipliers for Domains With Corners, Report HTKK-MAT-A141, Institute of Mathematics, Helsinki University of Technology, 1979.

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65N30

Retrieve articles in all journals with MSC: 65N30

Additional Information

Article copyright: © Copyright 1980 American Mathematical Society

American Mathematical Society