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An introduction to finite element methods for transient advection problems. (English) Zbl 0772.76035
This paper provides an excellent introduction to accurate finite element methods recently developed for solving unsteady problems governed by advection equations and hyperbolic equations, including explicit use of characteristic curves methods, methods based on the Taylor series in time, and least square methods. Essential informations on the accuracy and stability properties of the schemes are presented. This text can be considered as an easy introduction for those intending to solve practical problems. Some conclusions and an indication of areas requiring further research are also done.

MSC:
76M10 Finite element methods applied to problems in fluid mechanics
76R99 Diffusion and convection
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[1] Carey, G.F.; Oden, J.T., Finite elements: fluid mechanics, (1986), Prentice Hall Englewood Cliffs, NJ
[2] Johnson, C., Numerical solution of partial differential equations by the finite element method, (1987), Cambridge Univ. Press Cambridge
[3] Pironneau, O., Méthodes des eléments finis pour LES fluides, (1988), Masson Paris · Zbl 0748.76003
[4] John, F., Partial differential equations, (1982), Springer New York
[5] Quartapelle, L.; Rebay, S., Numerical solution of two-point boundary value problems, J. comput. phys., 86, 314-354, (1990) · Zbl 0692.65040
[6] Pironneau, O., On the transport-diffusion algorithm and its application to the Navier-Stokes equations, Numer. math., 38, 309-332, (1982) · Zbl 0505.76100
[7] Robert, A., A stable numerical integration scheme for the primitive meteorological equations, Atmos. Ocean, 19, 35-46, (1981)
[8] Robert, A., A semi-Lagrangian and semi-implicit numerical integration scheme for the primitive meteorological equations, J. meteorol. soc. (jpn.), 60, 319-325, (1982)
[9] Pudykiewicz, J.; Staniforth, A., Some properties and comparative performance of semi-Lagrangian method of robert in the solution of the advection-diffusion equation, Atmos. Ocean, 22, 283-308, (1984)
[10] Staniforth, A.; Pudykiewicz, J., Reply to comments on the addenda to “some properties and comparative performance of semi-Lagrangian method of robert in the solution of the advection-diffusion equation”, Atmos. Ocean, 23, 195-200, (1985)
[11] Tanguay, M.; Simard, A.; Staniforth, A., A three-dimensional semi-Lagrangian integration scheme for the Canadian regional finite-element forecast model, Monthly weather rev., 117, 1861-1871, (1989)
[12] Staniforth, A.; Côté, J., Semi-Lagrangian integration schemes and their application to environmental flows, (), 63-79
[13] Purnell, D.K., Solution of the advective equation by upstream interpolation with a cubic spline, Monthly weather rev., 104, 42-48, (1976)
[14] Benqué, J.P.; Ronat, J., Quelques difficultés des modeles numeriques en hydraulique, () · Zbl 0505.76035
[15] Benqué, J.P.; Ibler, B.; Keramsi, A.; Labadie, G., A new finite element method for the Navier-Stokes equations coupled with a temperature equation, (), 295-301 · Zbl 0457.76023
[16] Strang, G.; Fix, G.J., An analysis of the finite element method, (1973), Prentice Hall Englewood Cliffs, NJ · Zbl 0278.65116
[17] Morton, K.W., Generalized Galerkin methods for hyperbolic problems, Comput. methods appl. mech. engrg., 52, 847-871, (1985) · Zbl 0568.76007
[18] Morton, K.W.; Süli, E.E.; Priestly, A., A stability analysis of the Lagrange-Galerkin method with non-exact integration, Oxford univ. comput. lab., N.A. report 14, (1986)
[19] Morton, K.W., Finite element methods for hyperbolic equations, () · Zbl 0451.76001
[20] Morton, K.W.; Parrott, A.K., Generalized Galerkin methods for first-order hyperbolic equations, J. comput. phys., 36, 249-270, (1980) · Zbl 0458.65098
[21] Swartz, B.; Wendroff, B., The relation between the Galerkin and collocation methods using smooth splines, SIAM J. number. anal., 11, 1059-1068, (1974)
[22] Cullen, M.J.P.; Morton, K.W., Analysis of evolutionary error in finite element and other methods, J. comput. phys., 34, 245-267, (1980) · Zbl 0477.65064
[23] Donea, J., A Taylor-Galerkin method for convective transport problems, Internat. J. numer. methods engrg., 20, 101-120, (1984) · Zbl 0524.65071
[24] Donea, J.; Giuliani, S.; Laval, H.; Quartapelle, L., Time-accurate solution of advection-diffusion problems by finite elements, Comput. methods appl. mech. engrg., 45, 123-145, (1984) · Zbl 0514.76083
[25] Selmin, V.; Donea, J.; Quartapelle, L., Finite element methods for nonlinear advection, Comput. methods appl. mech. engrg., 52, 817-845, (1985) · Zbl 0573.76005
[26] Donea, J.; Quartapelle, L.; Selmin, V., An analysis of time discretization in the finite element solution of hyperbolic problems, J. comput. phys., 70, 463-499, (1987) · Zbl 0621.65102
[27] Selmin, V., Third-order finite element schemes for the solution of hyperbolic problems, INRIA report 707, (1987)
[28] Laval, H., Taylor-Galerkin solution of the time-dependent Navier-Stokes equations, (), 414-421
[29] Laval, H.; Quartapelle, L., A fractional-step Taylor-Galerkin method for unsteady incompressible flows, Internat. J. numer. methods fluids, 11, 501-513, (1990) · Zbl 0711.76019
[30] Hughes, T.J.R.; Mallet, M., A new finite element formulation for computational fluid dynamics: III. the generalized streamline operator for multidimensional advective-diffusive systems, Comput. methods appl. mech. engrg., 58, 305-328, (1986) · Zbl 0622.76075
[31] Donea, J.; Selmin, V.; Quartapelle, L., Recent developments of the Taylor-Galerkin method for the numerical solution of hyperbolic problems, (), 171-185 · Zbl 0661.76068
[32] Löhner, R.; Morgan, K.; Zienkiewicz, O.C., The solution of nonlinear hyperbolic equation systems by the finite element method, Internat. J. numer. methods fluids, 4, 1043-1063, (1984) · Zbl 0551.76002
[33] Oden, J.T.; Strouboulis, T.; Devloo, P., Adaptive finite element methods for the analysis of inviscid compressible flow: I. fast refinement/unrefinement and moving mesh methods for unstructured meshes, Comput. methods appl. mech. engrg., 59, 327-362, (1986) · Zbl 0593.76080
[34] Oden, J.T.; Strouboulis, T.; Devloo, P., Adaptive finite element methods for high-speed compressible flows, Internat. J. numer. methods fluids, 7, 1211-1228, (1987)
[35] Argyris, J.; St. Doltsinis, I.; Friz, H., Hermes space shuttle: exploration of reentry aerodynamics, Comput. methods appl. mech. engrg., 73, 1-51, (1989) · Zbl 0676.76058
[36] Harten, A.; Tal-Ezer, H., On fourth order accurate implicit finite difference scheme for hyperbolic conservation laws: I. nonstiff strongly dynamic problems, Math. comput., 36, 335-373, (1981) · Zbl 0468.65051
[37] Harten, A.; Tal-Ezer, H., On fourth order accurate implicit finite difference scheme for hyperbolic conservation laws: II. five-point schems, J. comput. phys., 41, 329-356, (1981) · Zbl 0468.65052
[38] Bristeau, M.O.; Pironneau, O.; Glowinski, R.; Periaux, J.; Perrier, P., On the numerical solution of nonlinear problems of fluid dynamics by least square and finite element methods — part I. least squares formulations and conjugate gradient solutions of the continuous problems, Comput. methods appl. mech. engrg., 17/18, 619-657, (1979) · Zbl 0423.76047
[39] Glowinski, R., Numerical methods for nonlinear variational problems, (1984), Springer New York · Zbl 0575.65123
[40] Li, C.W., Least square-characteristics and finite elements for advection-dispersion simulation, Internat J. numer. methods engrg., 29, 1343-1364, (1990) · Zbl 0728.73055
[41] Carey, G.F.; Jiang, B.N., Least-squares finite elements for first-order hyperbolic systems, Internat. J. numer. methods engrg., 26, 81-93, (1988) · Zbl 0641.65080
[42] Park, N.-S.; Liggett, J.A., Taylor-least squares finite element for two-dimensional advection dominated advection-diffusion problems, Internat. J. numer. methods fluids, 11, 21-38, (1990) · Zbl 0696.76105
[43] Park, N.-S.; Liggett, J.A., Application of Taylor-least squares finite element to three dimensional advection-diffusion problems, Internat. J. numer. methods fluids, 13, 759-773, (1991) · Zbl 0739.76033
[44] Nguyen, H.; Reynen, J., A space-time least-squares finite element scheme for advection-diffusion equations, Comput. methods appl. mech. engrg., 42, 331-342, (1984) · Zbl 0517.76089
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