## Equal-order interpolations: A unified approach to stabilize the incompressible and advective effects.(English)Zbl 0898.76068

We present a SUPG formulation for compressible and near incompressible Navier-Stokes equations. It introduces an extension of the exact solution for one-dimensional systems to the multidimensional case, in a similar way to that arising in the scalar problem. This formulation satisfies both the one-dimensional advective-diffusive system limit case and the advection-dominated multidimensional system case. Another feature of this formulation is that it introduces naturally a stabilizing term for the incompressibility condition. However, in our formulation the stabilization is introduced to the whole system of equations, while other authors introduce a term to stabilize the incompressibility condition and another one for the advective term.

### MSC:

 76M10 Finite element methods applied to problems in fluid mechanics 76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics 76D05 Navier-Stokes equations for incompressible viscous fluids
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### References:

 [1] Hughes, T.J.R.; Franca, L.; Balestra, M., A new finite element formulation for CFD: V. circumventing the babuska-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 [2] Frey, S.L.; Franca, L.P.; Sampaio, R., Stabilized methods for the incompressible Navier-Stokes flow, () [3] Zienkiewicz, O.C.; Szmelter, J.; Peraire, J., Compressible and incompressible flow: an algorithm for all seasons, Comput. methods appl. mech. engrg., 78, 105-121, (1990) · Zbl 0708.76099 [4] Zienkiewicz, O.C.; Wu, J., Incompressibility without tearsâ€”how to avoid restrictions on mixed formulations, Int. J. numer. methods engrg., 32, 1189-1204, (1991/1992) · Zbl 0756.76056 [5] Storti, M.; Nigro, N.; Idelsohn, S., Stabilizing equal-order interpolations for mixed formulations of Navier-Stokes equations, (), 194-202 · Zbl 0875.76300 [6] Hughes, T.J.R.; Tezduyar, T.E., Finite element methods for first-order hyperbolic systems with particular emphasis on the compressible Euler equations, Comput. methods appl. mech. engrg., 45, 217-284, (1984) · Zbl 0542.76093 [7] Hughes, T.J.R.; Mallet, M.; Mizukami, A., A new finite element method for CFD: II. beyond SUPG, Comput. methods appl. mech. engrg., 54, 341-355, (1986) · Zbl 0622.76074 [8] Hughes, T.J.R.; Mallet, M., A new finite element method for CFD: IV. A discontinuity-capturing operator for multidimensional advective-diffusive systems, Comput. methods appl. mech. engrg., 58, 329-336, (1986) · Zbl 0587.76120 [9] Mallet, M., A finite element method for CFD, () [10] Baiocchi, C.; Brezzi, F.; Franca, L., Virtual bubbles and Galerkin-least-squares type methods (ga.L.S.), Comput. methods appl. mech. engrg., 105, 125-141, (1993) · Zbl 0772.76033 [11] Baumann, C.; Storti, M.; Idelsohn, S., A Petrov-Galerkin technique for the solution of transonic and supersonic flows, Comput. methods appl. mech. engrg., 95, 49-70, (1992) · Zbl 0757.76024 [12] Storti, M.; Baumann, C.; Idelsohn, S., A preconditioning mass matrix to accelerate the convergence to the steady Euler solutions using explicit schemes, Int. J. numer. methods engrg., 34, 519-541, (1992) · Zbl 0772.76054 [13] Schreiber, R.; Keller, H.P., Driven cavity flows by efficient numerical techniques, J. comput. phys., 49, 310-333, (1983) · Zbl 0503.76040 [14] Lippke, A.; Wagner, H., Numerical solution of the Navier-Stokes equations in multiply domains connected, Comput. fluids, 20, 1, 19-27, (1991) · Zbl 0729.76060
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