Stabilized finite element methods. I.: Application to the advective- diffusive model. (English) Zbl 0759.76040

Summary: Some stabilized finite element methods for the Stokes problem are reviewed. The Douglas-Wang approach [J. Douglas jun. and J. Wang, Math. Comput. 52, No. 186, 495-508 (1989; Zbl 0669.76051)] confirms better stability features for high order interpolations. Next, the advective-diffusive model is approximated in the light of various stabilized methods, a global convergence analysis is presented and numerical experiments are performed. Biquadratic elements produce better numerical results under all stabilized methods examined. The design of the stability parameter is confirmed to be a crucial ingredient for simulating the advective-diffusive model, and some improved possibilities are suggested. Combinations of these methodologies are given in the conclusions and will be examined in detail in the sequel to this paper applied to the incompressible Navier-Stokes equations.


76M10 Finite element methods applied to problems in fluid mechanics
76D07 Stokes and related (Oseen, etc.) flows
76R99 Diffusion and convection


Zbl 0669.76051
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[1] Brooks, A.N.; Hughes, T.J.R., Streamline upwind/Petrov-Galerkin methods for advection dominated flows, () · Zbl 0449.76077
[2] Brooks, A.N.; Hughes, T.J.R., Streamline upwind/Petrov-Galerkin formulations for convective dominated flows with particular emphasis on the incompressible Navier-Stokes equations, Comput. methods appl. mech. engrg., 32, 199-259, (1982) · Zbl 0497.76041
[3] Hughes, T.J.R.; Brooks, A.N., A multidimensional upwind scheme with no crosswind diffusion, (), 19-35 · Zbl 0423.76067
[4] Hughes, T.J.R.; Brooks, A.N., A theoretical framework for Petrov-Galerkin methods with discontinuous weighting functions: application to the streamline upwind procedure, (), 46-65
[5] Johnson, C.; Nävert, U., An analysis of some finite element methods for advection-diffusion problems, (), 99-116
[6] Nävert, U., A finite element method for convection-diffusion problem, ()
[7] Hughes, T.J.R.; Mallet, M.; Franca, L.P., Entropy-stable finite element methods for compressible fluids: application to high order Mach number flows with shocks, (), 761-773
[8] Hughes, T.J.R.; Mallet, M.; Mizukami, A., A new finite element method for computational fluid dynamics: II. beyond SUPG, Comput. methods appl. mech. engrg., 54, 341-355, (1986) · Zbl 0622.76074
[9] Hughes, T.J.R.; Shakib, F., Computational aerodynamics and the finite element method, ()
[10] Johnson, C., Finite element methods for convection-diffusion problems, (), 311-323
[11] Johnson, C., Numerical solution of partial differential equations by the finite element method, (1987), Studentlitteratur Sweden
[12] Johnson, C.; Saranen, J., Streamline diffusion methods for the incompressible Euler and Navier-Stokes equations, Math. comp., 47, 1-18, (1986) · Zbl 0609.76020
[13] Johnson, C.; Nävert, U.; Pitkäranta, J., Finite element methods for linear hyperbolic problem, Comput. methods appl. mech. engrg., 45, 285-312, (1984) · Zbl 0526.76087
[14] Johnson, C.; Szepessy, A.; Hansbo, P., On the convergence of shock-capturing streamline diffusion finite element methods for hyperbolic conservation laws, Math. comp., 54, 107-129, (1990) · Zbl 0685.65086
[15] Szepessy, A., Convergence of the streamline diffusion finite element method for conservation laws, () · Zbl 0751.65061
[16] Hughes, T.J.R.; Franca, L.P.; Hulbert, G.M., A new finite element formulation for computational fluid dynamics: VIII. the Galerkin/least-squares method for advective-diffusive equations, Comput. methods appl. mech. engrg., 73, 173-189, (1989) · Zbl 0697.76100
[17] 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
[18] 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
[19] Douglas, J.; Wang, J., An absolutely stabilized finite element method for the Stokes problem, Math. comp., 52, 495-508, (1989) · Zbl 0669.76051
[20] Franca, L.P.; Stenberg, R., Error analysis of some Galerkin-least-squares methods for the elasticity equations, () · Zbl 0759.73055
[21] Dutra do Carmo, E.G.; Galeão, A.C., High order P-G finite elements for convection-dominated problems, (), 151-156
[22] Galeão, A.C.; Dutra do Carmo, E.G., A consistent approximate upwind Petrov-Galerkin method for convection-dominated problems, Comput. methods appl. mech. engrg., 68, 83-95, (1988) · Zbl 0626.76091
[23] Harari, I., Computational methods for problems of acoustics with particular reference to exterior domains, ()
[24] 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 formulations that converge for all velocity/pressure spaces, Comput. methods appl. mech. engrg., 65, 85-96, (1987) · Zbl 0635.76067
[25] 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
[26] Brezzi, F.; Pitkäranta, J., On the stabilization of finite element approximations of the Stokes problem, (), 11-19 · Zbl 0552.76002
[27] Kechkar, N.; Silvester, D.J., The stabilization of low order mixed finite element methods for incompressible flows, (), 111-116
[28] Ciarlet, P.G., The finite element method for elliptic problems, (1978), North-Holland Amsterdam · Zbl 0445.73043
[29] P. Hansbo and A. Szepessy, A velocity-pressure streamline diffusion finite element method for the incompressible Navier-Stokes equations, Preprint. · Zbl 0716.76048
[30] Tezduyar, T.E.; Shih, R.; Mittal, S.; Ray, S.E., Incompressible flow using stabilized bilinear and linear equal-order-interpolation velocity-pressure elements, () · Zbl 0756.76048
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