Virtual bubbles and Galerkin-least-squares type methods (Ga.L.S.).(English)Zbl 0772.76033

Summary: The equivalence between stabilized finite element methods (or Galerkin- least-squares type methods) and the standard Galerkin method with bubble functions is established in an abstract framework. The results are applicable to various finite element spaces, including high order elements, and applications to the advective diffusive model and to the Stokes problem are presented.

MSC:

 76M10 Finite element methods applied to problems in fluid mechanics 76M30 Variational methods applied to problems in fluid mechanics 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
Full Text:

References:

 [1] Brezzi, F.; Pitkäranta, J., On the stabilization of finite element approximations of the Stokes problem, (Hackbusch, W., Efficient Solutions of Elliptic Systems, Notes on Numerical Fluid Mechanics, Vol. 10 (1984), Vieweg: Vieweg Wiesbaden), 11-19 · Zbl 0552.76002 [2] Brezzi, F.; Douglas, J., Stabilized mixed methods for Stokes problem, Numer. Math., 53, 225-236 (1988) · Zbl 0669.76052 [3] Douglas, J.; Wang, J., An absolutely stabilized finite element method for the Stokes problem, Math. Comp., 52, 495-508 (1989) · Zbl 0669.76051 [4] 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 [5] Franca, L. P.; Stenberg, R., Error analysis of some Galerkin-least-squares methods for the elasticity equations, SIAM J. Numer. Anal., 28, 1680-1697 (1991) · Zbl 0759.73055 [6] 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 formulation that converge for all velocity/pressure spaces, Comput. Methods Appl. Mech. Engrg., 65, 85-96 (1987) · Zbl 0635.76067 [7] 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 [8] Johnson, C., Numerical Solution of Partial Differential Equations by the Finite Element Method (1987), Cambridge Univ. Press: Cambridge Univ. Press Cambridge [9] Arnold, D. N.; Brezzi, F.; Fortin, M., A stable finite element for the Stokes equations, Calcolo, 23, 4, 337-344 (1984) · Zbl 0593.76039 [10] Brezzi, F.; Bristeau, M.-O.; Franca, L. P.; Mallett, M.; Rogé, G., A relationship between stabilized finite element methods and the Galerkin method with bubble functions, Comput. Methods Appl. Mech. Engrg., 96, 117-129 (1992) · Zbl 0756.76044 [11] Rogé, G., On the approximation and the convergence acceleration in the finite element numerical simulation of compressible viscous flows, (Thèse de Doctorat (1990), Université Paris VI) [12] Pierre, R., Simple $$C^0$$ approximations for the computation of incompressible flows, Comput. Methods Appl. Mech. Engrg., 68, 205-227 (1988) · Zbl 0628.76040 [13] Arnold, D. N., Innovative finite element methods for plates, Mat. Apl. Comput., 10, 77-88 (1991) · Zbl 0757.73048 [14] Franca, L. P.; Frey, S. L.; Hughes, T. J.R., Stabilized finite element methods: I. Application to the advective-diffusive model, Comput. Methods Appl. Mech. Engrg., 95, 253-276 (1992) · Zbl 0759.76040
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.