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Hughes, Thomas J. R.; Franca, Leopoldo P.; Hulbert, Gregory M.

A new finite element formulation for computational fluid dynamics. VIII. The Galerkin/least-squares method for advective-diffusive equations. (English) Zbl 0697.76100

Comput. Methods Appl. Mech. Eng. 73, No. 2, 173-189 (1989).
Summary: [For part VII, see the authors, ibid. 65, 85-96 (1987; Zbl 0635.76067).]
Galerkin/least-squares finite element methods are presented for advective-diffusive equations. Galerkin/least-squares represents a conceptual simplification of SUPG, and is in fact applicable to a wide variety of other problem types. A convergence analysis and error estimates are presented.

Cited in 2 Reviews
Cited in 517 Documents

MSC:

76R50 Diffusion
65Z05 Applications to the sciences
76M99 Basic methods in fluid mechanics

Keywords:

streamline-diffusion methods; boundary value problems; scalar steady advection-diffusion equation; Galerkin/least-squares finite element methods; advective-diffusive equations; SUPG; convergence analysis; error estimates

Citations:

Zbl 0635.76067
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\textit{T. J. R. Hughes} et al., Comput. Methods Appl. Mech. Eng. 73, No. 2, 173--189 (1989; Zbl 0697.76100)
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References:

[1] Ciarlet, P.G., The finite element method for elliptic problems, (1978), North-Holland Amsterdam · Zbl 0445.73043
[2] Franca, L.P.; Hughes, T.J.R., Two classes of mixed finite element methods, Comput. meths. appl. mech. engrg., 69, 89-129, (1988) · Zbl 0651.65078
[3] 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
[4] Hughes, T.J.R.; Franca, L.P.; Harari, I.; Mallet, M.; Shakib, F.; Spelce, T.E., Finite element method for high-speed flows: consistent calculation of boundary flux, ()
[5] Hughes, T.J.R., The finite element method: linear static and dynamic finite element analysis, (1987), Prentice-Hall Englewood Cliffs, NJ
[6] 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. meths. fluids, 7, 1261-1275, (1987) · Zbl 0638.76080
[7] Hughes, T.J.R.; Franca, L.P., A new finite element method for computational fluid dynamics: VII. the Stokes problem with various well-posed boundary conditions: symmetric formulations that converge for all velocity/pressure spaces, Comput. meths. appl. mech. engrg., 65, 85-96, (1987) · Zbl 0635.76067
[8] Hughes, T.J.R.; Franca, L.P., A mixed finite element formulation for Reissner-Mindlin plate theory: uniform convergence of all high-order spaces, Comput. meths. appl. mech. engrg., 67, 223-240, (1988) · Zbl 0611.73077
[9] Hughes, T.J.R.; Franca, L.P.; Mallet, M., A new finite element formulation for computational fluid dynamics: VI. convergence analysis of the generalized SUPG formulation for linear time-dependent multidimensional advective-diffusive systems, Comput. meths. appl. mech. engrg., 63, 97-112, (1987) · Zbl 0635.76066
[10] Hughes, T.J.R.; Hulbert, G.M., Space-time finite element methods for elastodynamics: formulations and error estimates, Comput. meths. appl. mech. engrg., 66, 339-363, (1988) · Zbl 0616.73063
[11] Hughes, T.J.R.; Mallet, M., A new finite element formulation for computational fluid dynamics: III. the generalized streamline operator for multidimensional advection-diffusion systems, Comput. meths. appl. mech. engrg., 58, 305-328, (1986) · Zbl 0622.76075
[12] Hughes, T.J.R.; Mallet, M., A new finite element formulation for computational fluid dynamics: IV. A discontinuity-capturing operator for multidimensional advective-diffusive systems, Comput. meths. appl. mech. engrg., 58, 329-336, (1986) · Zbl 0587.76120
[13] Hughes, T.J.R.; Mallet, M.; Mizukami, A., A new finite element formulation for computational fluid dynamics: II. beyond SUPG, Comput. meths. appl. mech. engrg., 54, 341-355, (1986) · Zbl 0622.76074
[14] Johnson, C., Streamline diffusion methods for problems in fluid mechanics, (), 251-261
[15] Johnson, C., Numerical solutions of partial differential equations by the finite element method, (1987), Cambridge University Press Cambridge
[16] Johnson, C.; Nävert, U.; Pitkäranta, J., Finite element methods for linear hyperbolic problems, Comput. meths. appl. mech. engrg., 45, 285-312, (1984) · Zbl 0526.76087
[17] Johnson, C.; Szepessy, A., On the convergence of streamline diffusion finite element methods for hyperbolic conservation laws, () · Zbl 0685.65086
[18] Johnson, C.; Szepessy, A.; Hansbo, P., On the convergence of shock-capturing streamline diffusion finite element methods for hyperbolic conservation laws, () · Zbl 0685.65086
[19] 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. meths. appl. mech. engrg., 63, 281-303, (1987) · Zbl 0607.73077
[20] Loula, A.F.D.; Hughes, T.J.R.; Franca, L.P.; Miranda, I., Mixed Petrov-Galerkin methods for the Timoshenko beam, Comput. meths. appl. mech. engrg., 63, 133-154, (1987) · Zbl 0607.73076
[21] Loula, A.F.D.; Miranda, I.; Hughes, T.J.R.; Franca, L.P., A successful mixed formulation for axisymmetric shell analysis employing discontinuous stress field of the same order as the displacement field, (), 581-599, Salvador, Brazil
[22] Nävert, U., A finite element method for convection-diffusion problems, ()
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