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A Douglas-Wang finite element approach for transient advection-diffusion problems. (English) Zbl 0845.76044
The paper presents an effective finite element method for approximating the two-dimensional unsteady advection-diffusion transport. The new scheme is a direct generalization of the Douglas-Wang finite element method applied to steady advection-diffusion and Stokes problems. Higher-order spatial derivatives in the new formulation necessitate higher-degree polynomial shape function. Quadratic finite element approximation (quadrilateral $$Q9$$ and triangular $$T6$$) are used. The finite difference $$\theta$$-weighting formulation is used for a time discretization. A stability analysis, using the von Neumann method, shows that the formulation based on the implicit Crank-Nicholson scheme reduces dissipation and dispersion errors when both advective and diffusive problems are considered.

MSC:
 76M10 Finite element methods applied to problems in fluid mechanics 76R50 Diffusion
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References:
 [1] Tabata, M., A finite element approximation corresponding to the upwind finite differencing, Mem. numer. math., 4, 47-63, (1977) · Zbl 0358.65102 [2] Baba, K.; Tabat, M., On a conservative upwind finite element scheme for convective diffusion equations, RAIRO numer. anal., 15, 3-25, (1981) · Zbl 0466.76090 [3] Kanayama, H., Discrete models for salinity distribution in a bay: conservation law and maximum principle, Theor. appl. mech., 28, 559-579, (1978) [4] Ikeda, T., Artificial viscosity in finite element approximations to the diffusion equation with drift terms, Lecture notes numer. appl. anal., 2, 59-78, (1980) [5] Ikeda, T., Maximum principle in finite element models for convection-diffusion phenomena, (1983), Kinokuniya Tokyo · Zbl 0508.65049 [6] Leonard, B.P., A survey of finite differences of opinion on numerical muddling of the incomprehensible defective confusion equation, () · Zbl 0435.76003 [7] 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 [8] Kelly, D.W.; Nakazawa, S.; Zienkiewicz, O.C., A note on upwinding and anisotropic balancing dissipation in finite element approximations to convective diffusion problem, Internat. J. numer. methods engrg., 15, 1705-1711, (1980) · Zbl 0452.76068 [9] Mizukami, A.; Hughes, T.J.R., A Petrov-Galerkin finite element method for convection-dominated flows: an accurate upwinding technique for satisfying the maximum principle, Comput. methods appl. mech. engrg., 50, 181-193, (1985) · Zbl 0553.76075 [10] Do Carmo, E.G.Dutra; Galeão, A.C., Feedback Petrov-Galerkin methods for convection-dominated problems, Comput. methods appl. mech. engrg., 88, 1-16, (1991) · Zbl 0753.76093 [11] Hughes, T.J.R.; Mallet, M.; Mizukami, A., A new finite element formulation for computational fluid dynamics: II. beyond SUPG, Comput. methods appl. mech. engrg., 54, 341-355, (1986) · Zbl 0622.76074 [12] Mallet, M., () [13] Galeão, A.C.; Do Carmo, E.G.Dutra, A consistent approximate upwind Petrov-Galerkin method for convection-dominated problems, Comput. methods appl. mech. engrg., 68, 83-95, (1988) · Zbl 0626.76091 [14] 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. methods appl. mech. engrg., 58, 329-336, (1986) · Zbl 0587.76120 [15] 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 [16] Donea, J., A Taylor-Galerkin method for convective transport problems, Internat. J. numer. methods engrg., 20, 101-119, (1984) · Zbl 0524.65071 [17] Donea, J.; Quartapelle, L.; Selmin, V., An analysis of time discretization in finite element solution of hyperbolic problems, J. comput. phys., 70, 463-499, (1987) · Zbl 0621.65102 [18] Carey, G.F.; Jiang, B.N., Least-squares finite element for first-order hyperbolic systems, Internat. J. numer. methods engrg., 26, 81-93, (1988) · Zbl 0641.65080 [19] 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 [20] Jiang, B.-N.; Povinelli, L.A., Least-squares finite element method for fluid dynamics, Comput. methods appl. mech. engrg., 81, 13-37, (1990) · Zbl 0714.76058 [21] Shakib, F.; Hughes, T.J.R., A new finite element formulation for computational fluid dynamics: IX. Fourier analysis of space-time Galerkin/least-squares algorithms, Comput. methods appl. mech. engrg., 87, 35-58, (1991) · Zbl 0760.76051 [22] Park, N.S.; Liggett, J.A., Taylor-least-squares finite element for two dimensional advection-diffusion problems, Internat. J. methods fluids, 11, 21-38, (1990) · Zbl 0696.76105 [23] Douglas, J.; Wang, J., An absolutely stabilized finite element method for the Stokes problem, Math. comp., 485-508, (1989) · Zbl 0669.76051 [24] 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 [25] Harari, I., Computational methods for problems of acoustics with particular reference to exterior domains, () [26] Khelifa, A., Nouvelle approche en éléments finis pour la modélisation du phénomène de transport permanent et non-permanent, () [27] Hirsch, C., Numerical computation of internal and external flows, (1988), Wiley New York [28] Morton, K.W.; Parrot, A.K., Generalized Galerkin methods for first-order hyperbolic equations, J. comput. phys., 36, 249-270, (1980) · Zbl 0458.65098 [29] Bezier, F., Problèmes de transport-diffusion par éléments finis, () [30] Frenette, R., REFSED-3D: modélisation tridimensionnelle par éléments finis du transport de sédiments par diffusion et convection, () [31] Bouloutas, E.T.; Celia, M.A., An improved cubic Petrov-Galerkin method for simulation of transient advection-diffusion processes in rectangularly decomposable domains, Comput. methods appl. mech. engrg., 92, 289-308, (1991) · Zbl 0825.76609 [32] Codina, R.; Oñate, E.; Cervera, M., The intrinsic time for the streamline upwind/Petrov-Galerkin formulation using quadratic elements, Comput. methods appl. mech. engrg., 94, 239-262, (1992) · Zbl 0748.76082
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