×

zbMATH — the first resource for mathematics

On a two-level finite element method for the incompressible Navier-Stokes equations. (English) Zbl 1002.76066
Summary: We consider Galerkin finite element method for incompressible Navier-Stokes equations in two dimensions, where the finite-dimensional space(s) employed consist of piecewise polynomials enriched with residual-free bubble functions. To find the bubble part of the solution, a two-level finite element method (TLFEM) is described, and its application to Navier-Stokes equation is displayed. Numerical solutions employing TLFEM are presented for three benchmark problems. We compare numerical solutions obtained by using TLFEM with numerical solutions obtained by using a stabilized method.

MSC:
76M10 Finite element methods applied to problems in fluid mechanics
76D05 Navier-Stokes equations for incompressible viscous fluids
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms, Springer Series in Computational Mathematics, vol. 5. Springer: Berlin, New-York, 1986. · Zbl 0585.65077 · doi:10.1007/978-3-642-61623-5
[2] (eds). Incompressible Computational Fluid Dynamics: Trends and Advances. Cambridge University Press: Cambridge, New York, 1993. · Zbl 1190.76004 · doi:10.1017/CBO9780511574856
[3] (eds). Numerical Methods for the Navier-Stokes Equations, Notes on Numerical Fluid Mechanics, vol. 47. Vieweg: Braunschweig, Wiesbaden, 1994. · doi:10.1007/978-3-663-14007-8
[4] Oden, AIAA Journal 10 pp 1590– (1972) · Zbl 0254.76043 · doi:10.2514/3.6691
[5] Taylor, Computational Fluids 1 pp 73– (1973) · Zbl 0328.76020 · doi:10.1016/0045-7930(73)90027-3
[6] Brezzi, Numerische Mathematik 53 pp 225– (1988) · Zbl 0669.76052 · doi:10.1007/BF01395886
[7] On the stabilization of finite element approximations of the Stokes problem. In Efficient Solutions of Elliptic Systems, Notes on Numerical Fluid Mechanics, vol. 10, (ed). Vieweg: 1984; 11-19. · doi:10.1007/978-3-663-14169-3_2
[8] Douglas, Mathematics of Computation 52 pp 495– (1989) · doi:10.1090/S0025-5718-1989-0958871-X
[9] Franca, Computer Methods in Applied Mechanics and Engineering 69 pp 89– (1988) · Zbl 0629.73053 · doi:10.1016/0045-7825(88)90168-5
[10] Stabilized finite element methods for the Stokes problem. In Incompressible Computational Fluid Dynamics?Trends and Advances, (eds). Cambridge University Press: Cambridge, 1993; 87-107. · doi:10.1017/CBO9780511574856.005
[11] Hughes, Computer Methods in Applied Mechanics and Engineering 65 pp 85– (1987) · Zbl 0635.76067 · doi:10.1016/0045-7825(87)90184-8
[12] Hughes, Computer Methods in Applied Mechanics and Engineering 59 pp 85– (1986) · Zbl 0622.76077 · doi:10.1016/0045-7825(86)90025-3
[13] Arnold, Calcolo 23 pp 337– (1984) · Zbl 0593.76039 · doi:10.1007/BF02576171
[14] Babu?ka, Numerirsche Mathematik 20 pp 179– (1973) · Zbl 0258.65108 · doi:10.1007/BF01436561
[15] Brezzi, RAIRO Ser. Rouge 8 pp 129– (1974)
[16] Franca, Computer Methods in Applied Mechanics and Engineering 105 pp 395– (1993) · Zbl 0776.76049 · doi:10.1016/0045-7825(93)90065-6
[17] Baiocchi, Computer Methods in Applied Mechanics and Engineering 105 pp 125– (1993) · Zbl 0772.76033 · doi:10.1016/0045-7825(93)90119-I
[18] Brezzi, Computer Methods in Applied Mechanics and Engineering 96 pp 117– (1992) · Zbl 0756.76044 · doi:10.1016/0045-7825(92)90102-P
[19] Stabilization techniques and subgrid scales capturing. In Proceedings of the Conference ?State of the Art in Numerical Analysis?, York, England, April 1-4, 1996.
[20] Brezzi, b=?g. Computer Methods in Applied Mechanics and Engineering 145 pp 329– (1997) · Zbl 0904.76041 · doi:10.1016/S0045-7825(96)01221-2
[21] Franca, Computer Methods in Applied Mechanics and Engineering 123 pp 299– (1995) · Zbl 1067.76567 · doi:10.1016/0045-7825(94)00721-X
[22] Hughes, Computer Methods in Applied Mechanics and Engineering 127 pp 387– (1995) · Zbl 0866.76044 · doi:10.1016/0045-7825(95)00844-9
[23] Pierre, Computer Methods in Applied Mechanics and Engineering 68 pp 205– (1988) · Zbl 0628.76040 · doi:10.1016/0045-7825(88)90116-8
[24] Pierre, Numerical Methods for Partial Differential Equations 5 pp 241– (1989) · Zbl 0672.76038 · doi:10.1002/num.1690050307
[25] Bank, Computer Methods in Applied Mechanics and Engineering 83 pp 61– (1990) · Zbl 0732.65100 · doi:10.1016/0045-7825(90)90124-5
[26] Numerical Solution of Convection-Diffusion Problems. Chapman & Hall: London, 1996.
[27] Numerical Methods for Singularly Perturbed Differential Equations. Springer: Berlin, 1996. · doi:10.1007/978-3-662-03206-0
[28] Brooks, Computer Methods in Applied Mechanics and Engineering 32 pp 199– (1982) · Zbl 0497.76041 · doi:10.1016/0045-7825(82)90071-8
[29] Johnson, Mathematics of Computation 46 pp 1– (1986) · doi:10.1090/S0025-5718-1986-0815828-4
[30] Shishkin, Russian Journal on Numerical Analysis in Mathematical Modelling 7 pp 537– (1992)
[31] Cai, SIAM Journal on Numerical Analysis 31 pp 1785– (1994) · Zbl 0813.65119 · doi:10.1137/0731091
[32] Cai, SIAM Journal on Numerical Analysis 34 pp 425– (1997) · Zbl 0912.65089 · doi:10.1137/S0036142994266066
[33] Fiard, SIAM Journal on Scientific Computing 19 pp 1958– (1998) · Zbl 0911.65108 · doi:10.1137/S1064827596301169
[34] Brezzi, Computer Methods in Applied Mechanics and Engineering 166 pp 25– (1998) · Zbl 0934.65126 · doi:10.1016/S0045-7825(98)00080-2
[35] Brezzi, Mathematical Models and Methods in Applied Sciences 4 pp 571– (1994) · Zbl 0819.65128 · doi:10.1142/S0218202594000327
[36] Franca, Applied Mathematics Letters 9 pp 83– (1996) · Zbl 0903.65082 · doi:10.1016/0893-9659(96)00078-X
[37] Russo, Computer Methods in Applied Mechanics and Engineering 132 pp 335– (1996) · Zbl 0887.76038 · doi:10.1016/0045-7825(96)01020-1
[38] Approximating the incompressible Navier-Stokes equations using a two level finite element method. Ph.D. Thesis, University of Colorado, Denver, 1999.
[39] Russo, Applied Mathematics Letters 8 pp 1– (1995) · Zbl 0821.65066 · doi:10.1016/0893-9659(95)00001-7
[40] Russo, Mathematical Models and Methods in Applied Sciences 6 pp 33– (1996) · Zbl 0853.65109 · doi:10.1142/S0218202596000031
[41] Franca, International Journal for Numerical Methods in Engineering 43 pp 23– (1998) · Zbl 0935.65117 · doi:10.1002/(SICI)1097-0207(19980915)43:1<23::AID-NME383>3.0.CO;2-N
[42] Franca, Computer Methods in Applied Mechanics and Engineering 166 pp 35– (1998) · Zbl 0934.65127 · doi:10.1016/S0045-7825(98)00081-4
[43] Franca, Computer Methods in Applied Mechanics and Engineering 99 pp 209– (1992) · Zbl 0765.76048 · doi:10.1016/0045-7825(92)90041-H
[44] Tezduyar, Computer Methods in Applied Mechanics and Engineering 95 pp 221– (1992) · Zbl 0756.76048 · doi:10.1016/0045-7825(92)90141-6
[45] Sohn, International Journal for Numerical Methods in Fluids 8 pp 1469– (1988) · doi:10.1002/fld.1650081202
[46] Clustered element-by-element computations for fluid flow. In Parallel Computational Fluid Dynamics: Implementations and Results, (ed.). MIT Press: Cambridge, MA, 1990; 165-188.
[47] (eds). Analysis of Laminar Flow over a Backward Facing Step: A GAMM Workshop. Viewig: Wiesbaden, 1984. · doi:10.1007/978-3-663-14242-3
[48] Computing steady incompressible flows past blunt bodies: a historical overview. In Numerical Methods for Fluid Dynamics 4, (eds). Clarendon Press: London, 1993; 115-133. · Zbl 0802.76066
[49] An Album of Fluid Motion. The Parabolic Press: California, 1982.
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.