The finite element method with Lagrange multipliers on the boundary: Circumventing the Babuška-Brezzi condition. (English) Zbl 0764.73077
Summary: In order to circumvent the Babuška-Brezzi condition in the finite element method with Lagrange multipliers on the boundary, least-squares- like terms are added to the classical Galerkin method. The additional terms involve integrals over element interiors and mesh-parameter dependent coefficients. The resulting formulations retain consistency and attain convergence for arbitrary polynomial interpolations which are continuous for the primal variable and which may be continuous or discontinuous for the Lagrange multiplier.
MSC:
| 74S05 | Finite element methods applied to problems in solid mechanics |
| 74S30 | Other numerical methods in solid mechanics (MSC2010) |
| 74P10 | Optimization of other properties in solid mechanics |
References:
| [1] | Babuška, I., Error bounds for finite element method, Numer. Math., 16, 322-333 (1971) · Zbl 0214.42001 |
| [2] | Brezzi, F., On the existence, uniqueness and approximation of saddle-point problems arising from the Lagrange multipliers, Revue Française d’Automatique Informatique et Recherche Opérationelle, Ser. Rouge Anal. Numér., 8, R-2, 129-151 (1974) · Zbl 0338.90047 |
| [3] | Hughes, T. J.R., Recent progress in the development and understanding of SUPG methods with special reference to the compressible Euler and Navier-Stokes equations, Internat. J. Numer. Methods Fluids, 7, 1261-1275 (1987) · Zbl 0638.76080 |
| [4] | Hughes, T. J.R.; Franca, L. P.; Balestra, M., A new finite element formulation for computational fluid dynamics: V. Circumventing the Babuška-Brezzi condition: A stable Petrov-Galerkin formulation of the Stokes problem accommodating equal-order interpolations, Comput. Methods Appl. Mech. Engrg., 59, 85-99 (1986) · Zbl 0622.76077 |
| [5] | Hughes, T. J.R.; Franca, L. P., A new finite element formulation for computational fluid dynamics: VII. The Stokes problem with various well-posed boundary conditions: Symmetric formulatiosn that converge for all velocity/pressure spaces, Comput. Methods Appl. Mech. Engrg., 65, 85-96 (1987) · Zbl 0635.76067 |
| [6] | Loula, A. F.D.; Franca, L. P.; Hughes, T. J.R.; Miranda, I., Stability, convergence and accuracy of a new finite element method for the circular arch problem, Comput. Methods Appl. Mech. Engrg., 63, 281-303 (1987) · Zbl 0607.73077 |
| [7] | Loula, A. F.D.; Hughes, T. J.R.; Franca, L. P.; Miranda, I., Mixed Petrov-Galerkin methods for the Timoshenko beam, Comput. Methods Appl. Mech. Engrg., 63, 133-154 (1987) · Zbl 0607.73076 |
| [8] | Franca, L. P.; Hughes, T. J.R.; Loula, A. F.D.; Miranda, I., A new family of stable elements for nearly incompressible elasticity based on a mixed Petrov-Galerkin finite element formulation, Numer. Math., 53, 123-141 (1988) · Zbl 0656.73036 |
| [9] | Loula, A. F.D.; Miranda, I.; Hughes, T. J.R.; Franca, L. P., On mixed finite element methods for axisymmetric shell analysis, Comput. Methods Appl. Mech. Engrg., 72, 201-231 (1989) · Zbl 0691.73030 |
| [10] | Franca, L. P.; Hughes, T. J.R., Two classes of mixed finite element methods, Comput. Methods Appl. Mech. Engrg., 69, 89-129 (1988) · Zbl 0651.65078 |
| [11] | Lions, J. L.; Magenes, E., (Problèmes aux Limites Non Homogènes et Applications, Vol. 1 (1968), Dunod: Dunod Paris) · Zbl 0165.10801 |
| [12] | Babuška, I., The finite element method with Lagrange multipliers, Numer. Math., 20, 179-192 (1973) · Zbl 0258.65108 |
| [13] | Babuška, I.; Aziz, A. K., Survey lectures on the mathematical foundations of the finite element method, (Aziz, A. K., The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations (1972), Academic Press: Academic Press New York) · Zbl 0268.65052 |
| [14] | Pitkäranta, J., Boundary spaces for the finite element method with Lagrange multipliers, Numer. Math., 33, 273-289 (1979) · Zbl 0422.65062 |
| [15] | Hughes, T. J.R., The Finite Element Method: Linear Static and Dynamic Analysis (1987), Prentice-Hall: Prentice-Hall Englewood Cliffs, NJ · Zbl 0634.73056 |
| [16] | Franca, L. P.; Dutra do Carmo, E. G., The Galerkin gradient least-squares method, Comput. Methods Appl. Mech. Engrg., 74, 41-54 (1989) · Zbl 0699.65077 |
| [17] | Ciarlet, P. G., The Finite Element Method for Elliptic Problems (1978), North-Holland: North-Holland Amsterdam · Zbl 0445.73043 |
| [18] | Douglas, J.; Wang, J., An absolutely stabilized finite element method for the Stokes problems, Math. Comp., 52, 186, 495-508 (1989) · Zbl 0669.76051 |
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