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Stabilised bilinear-constant velocity-pressure finite elements for the conjugate gradient solution of the Stokes problem. (English) Zbl 0706.76075
Summary: In this paper, a sufficient condition for the stability of low-order mixed finite element methods is introduced. To illustrate the possibilities, two stabilization procedures for the popular $Q\sb 1-P\sb 0$ mixed method are theoretically analyzed. The effectiveness of these procedures in practice is assessed by comparing results with those obtained using a conventional penalty formulation, for a standard test problem. It is demonstrated that with appropriate stabilization, efficient iterative solution techniques of conjugate gradient type can be applied directly to the discrete Stokes system.

76M10Finite element methods (fluid mechanics)
76D07Stokes and related (Oseen, etc.) flows
Full Text: DOI
[1] Sani, R. L.; Gresho, P. M.; Lee, R. L.; Griffiths, D. F.: The cause and cure (?) of the spurious pressures generated by certain finite element method solutions of the incompressible Navier-Stokes equations, parts 1 and 2. Internat. J. Numer. methods fluids 1, 171-204 (1981) · Zbl 0461.76022
[2] Pitkäranta, J.; Saarinen, T.: A multigrid version of a simple finite element method for the Stokes problem. Math. comp. 45, 1-14 (1985) · Zbl 0584.65080
[3] M. Fortin and S. Boivin, Iterative stabilisation of the Bilinear velocity-constant pressure element, Internat. J. Numer. Methods Fluids, to appear. · Zbl 0687.76029
[4] Hughes, T. J. R.; Franca, L. P.: A new finite element formulation for CFD: VII. The Stokes problem with various well-posed boundary conditions: symmetric formulations that converge for all velocity/pressure spaces. Comput. methods appl. Mech. engrg. 65, 85-96 (1987) · Zbl 0635.76067
[5] Pironneau, O.: On the transport diffusion algorithm and its applications to the Navier-Stokes equations. Numer. math. 38, 309-332 (1982) · Zbl 0505.76100
[6] Brezzi, F.; Pitkäranta, J.: On the stabilisation of finite element approximations of the Stokes problem. Notes on numerical fluid mechanics 10, 11-19 (1984)
[7] Hughes, T. J. R.; Franca, L. P.; Balestra, M.: A new finite element formulation for CFD: 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] Brezzi, F.; Douglas, J.: Stabilized mixed methods for the Stokes problem. Numer. math. 53, 225-235 (1988) · Zbl 0669.76052
[9] Kechkar, N.: Ph.d. thesis. (1989)
[10] Silvester, D. J.; Thatcher, R. W.: The effect of the stability of mixed finite element approximations on the accuracy and rate of convergence of solution when solving incompressible flow problems. Internat. J. Numer. methods fluids 6, 841-853 (1986) · Zbl 0603.76025
[11] Boland, J. M.; Nicolaides, R. A.: On the stability of bilinear-constant velocity-pressure finite elements. Numer. math. 44, 219-222 (1984) · Zbl 0544.76030
[12] Pitkäranta, J.; Stenberg, R.: Error bounds for the approximation of the Stokes problem using bilinear/constant element on irregular quadrilateral meshes. The mathematics of finite elements and applications V, 325-334 (1984)
[13] Hughes, T. J. R.; Liu, W. K.; Brooks, A.: Finite element analysis of incompressible viscous flows by penalty function formulation. J. comput. Phys. 30, 1-60 (1979) · Zbl 0412.76023
[14] Johnson, C.; Pitkäranta, J.: Analysis of mixed finite element methods related to reduced integration. Math. comp. 42, 9-23 (1984) · Zbl 0482.65058
[15] Golub, G.; Van Loan, C.: Matrix computations. (1983) · Zbl 0559.65011
[16] Adams, L.: M-step preconditioned conjugate gradient methods. SIAM J. Sci. stat. Comput. 6, 452-463 (1985) · Zbl 0566.65018
[17] Ortega, J. M.; Voigt, R. G.: Solution of partial differential equations on vector and parallel computers. SIAM rev. 27, 149-240 (1985) · Zbl 0644.65075