Boundary Lagrange multipliers in finite element methods: Error analysis in natural norms. (English) Zbl 0765.65102

A finite element procedure for circumventing the Babuška-Brezzi condition in mixed formulations with Lagrange multipliers defined on the boundary is presented. The idea is to provide stability to the multiplier by adding residual terms constructed from the Euler-Lagrange equation to the classical Galerkin formulation while preserving consistency.
A symmetric conditionally stable method and a nonsymmetric absolutely stable one are shown to be consistent and converge in a natural mesh- independent norm. One order increase in the rate of convergence for the displacement in \(L_ 2\) is also obtained under standard assumptions on regularity.
Reviewer: V.Arnăutu (Iaşi)


65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
Full Text: DOI EuDML


[1] Babu?ka, I. (1971): Error bounds for finite element method. Numer. Math.16 322-333 · Zbl 0214.42001
[2] Babu?ka, I. (1973): The finite element method with Lagrange multipliers, Numer. Math.20, 179-192 · Zbl 0258.65108
[3] Barbosa, H.J.C., Hughes, T.J.R. (1991): The finite element method with Lagrange multipliers on the boundary: Circumventing the Babu?ka-Brezzi condition. Comput. Methods Appl. Mech. Eng.85 109-128 · Zbl 0764.73077
[4] Brezzi, F. (1974): On the existence, uniqueness and approximation of saddle-point problems arising from Lagrange multipliers. Rev. Fran?. Automatique Inform. Rech. Op?r., Ser. Rouge Anal. Num?r.8, R-2, 129-151
[5] Ciarlet, P.G. (1978): The Finite Element Method for Elliptic Problems. North Holland, Amsterdam · Zbl 0383.65058
[6] Franca, L.P., Hughes, T.J.R. (1988): Two classes of mixed finite element methods. Comput. Methods Appl. Mech. Eng.69, 89-129 · Zbl 0629.73053
[7] Hughes, T.J.R. (1987a): Recent progress in the development and understanding of SUPG methods with special reference to the compressible Euler and Navier-Stokes equations. Int. J. Numer. Methods Fluids7 1261-1275 · Zbl 0638.76080
[8] Hughes, T.J.R. (1987b): The Finite Element Method: Linear Static and Dynamic Analysis. Prentice Hall, Englewood Cliffs, N.J. · Zbl 0634.73056
[9] Lions, J.L., Magenes, E. (1968): Probl?mes aux limites non homog?nes et applications, Vol. 1. Dunod, Paris · Zbl 0165.10801
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