## Petrov-Galerkin solutions of the incompressible Navier-Stokes equations in primitive variables with adaptive remeshing.(English)Zbl 0783.76053

Summary: The solution of the incompressible Navier-Stokes equations in primitive variables is addressed. A Petrov-Galerkin formulation which automatically introduces streamline upwinding and allows equal order interpolation for all flow variables is presented. The solution algorithm is segregated, involving a series of updatings of the velocity and pressure fields. Error estimation and adaptive remeshing strategies for both steady and unsteady problems are demonstrated in some representative numerical examples.

### MSC:

 76M10 Finite element methods applied to problems in fluid mechanics 76M30 Variational methods applied to problems in fluid mechanics 76D05 Navier-Stokes equations for incompressible viscous fluids 65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs 65M50 Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs

### Keywords:

streamline upwinding; updatings; error estimation
Full Text:

### References:

 [1] Johnson, C., Numerical solutions of partial differential equations by the finite element method, (1987), Cambridge Univ. Press Cambridge [2] Morton, K.W., Generalised Galerkin methods for steady and unsteady problems, (), 1-32 · Zbl 0543.76118 [3] Heinrich, J.C.; Huyakorn, P.S.; Zienkiewicz, O.C.; Mitchell, A.R., An upwind finite element scheme for two-dimensional convective transport equations, Internat. J. numer. methods engrg., 11, 131-143, (1977) · Zbl 0353.65065 [4] Brooks, A.N.; Hughes, T.J.R., Streamline upwind/petrov—galerkin formulations for convection dominated flows with particular emphasis on the incompressible navier—stokes equations, Comput. methods appl. mech. engrg., 32, 199-259, (1982) · Zbl 0497.76041 [5] de Sampaio, P.A.B., A petrov—galerkin/modified operator formulation for convection-diffusion problems, Internat. J. numer. methods engrg., 30, 331-347, (1990) · Zbl 0714.76077 [6] Hughes, T.J.R.; Franca, L.P.; Ballestra, 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 [7] de Sampaio, P.A.B., A petrov—galerkin formulation for the incompressible navier—stokes equations using equal order interpolation for velocity and pressure, Internat. J. numer. methods engrg., 31, 1135-1149, (1991) · Zbl 0825.76430 [8] Jiang, B.N.; Carey, G.F., A stable least-squares finite element method for non-linear hyperbolic problems, Internat. J. numer. methods fluids, 8, 942-993, (1988) · Zbl 0666.76087 [9] 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 [10] de Sampaio, P.A.B., Petrov—galerkin finite element formulations for incompressible viscous flows, () · Zbl 1091.76042 [11] Patankar, S.V.; Spalding, D.B., A calculation procedure for heat, mass and momentum transfer in three-dimensional parabolic flows, Internat. J. heat mass transfer, 15, 1787-1806, (1972) · Zbl 0246.76080 [12] R.B. Willmersdorf, Private communication. [13] Lyra, P.R.M., An $$h- version$$ auto-adaptive FEM refinement strategy applied to two-dimensional field equation, (), (in Portuguese) [14] Zienkiewicz, O.C.; Zhu, J.Z., A simple error estimator and adaptive procedure for practical engineering analysis, Internat. J. numer. methods. engrg., 24, 337-357, (1987) · Zbl 0602.73063 [15] Wu, J.; Zhu, J.Z.; Szmelter, J.; Zienkiewicz, O.C., Error estimation and adaptivity in navier—stokes incompressible flow, () · Zbl 0699.76035 [16] Zienkiewicz, O.C.; Qu, S.; Taylor, R.L.; Nakazawa, S., The patch-test for mixed formulation, Internat. J. numer. methods. engrg., 23, 1873-1883, (1986) · Zbl 0614.65115 [17] Coutinho, A.L.G.A.; Alves, J.L.D.; Ebecken, N.F.F.; Troina, L.M., Conjugate gradient solution of finite element equations on the IBM 3090 vector computer utilizing polynomial preconditionings, Comput. methods. appl. mech. engrg., 84, 129-145, (1990) · Zbl 0729.73240 [18] Zienkiewicz, O.C.; Taylor, R.L., () [19] Szabó, B.; Babuška, I., Finite element analysis, (1991), Wiley New York [20] () [21] Weatherill, N.P., A method for generating irregular computational grids in multiply connected planar domains, Internat. J. numer. methods fluids, 8, 181-197, (1988) · Zbl 0641.76057 [22] Peraire, J.; Vahdati, M.; Morgan, K.; Zienkiewicz, O.C., Adaptive remeshing for compressible flow computations, J. comput. phys., 72, 449-466, (1987) · Zbl 0631.76085 [23] Weatherill, N.P., Mesh generation in computational fluid dynamics, () · Zbl 0761.76084 [24] Peraire, J., A finite element method for convection dominated flows, () [25] Lyra, P.R.M.; Alves, J.L.D.; Coutinho, A.L.G.A.; Landau, L.; Devloo, P.R.B., Comparison of mesh refinement strategies for the $$H- version$$ of the finite element method, (), A-595-A-610 [26] Lyra, P.R.M.; de Sampaio, P.A.B., Steady-state incompressible Navier-Stokes equations: solutions employing adaptive remeshing, (), (in Portuguese) · Zbl 0783.76053 [27] Burggraf, O.R., Analytical and numerical studies of the structure of steady separated flows, J. fluid mech., 24, 113-151, (1966) [28] Pironneau, O., Finite element methods for fluids, (1989), Wiley New York · Zbl 0665.73059 [29] Panton, R.C., Incompressible flow, (1984), Wiley New York · Zbl 0623.76001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.