# zbMATH — the first resource for mathematics

On a multi-scale element-free Galerkin method for the Stokes problem. (English) Zbl 1262.76026
Summary: A multi-scale element-free Galerkin method is presented for the Stokes problem. The new method is based on the Hughes’ variational multi-scale formulation, and arises from a decomposition of the velocity field into coarse/resolved scales and fine/unresolved scales. In this method, an unresolved model is obtained in which unresolved scales are incorporated analytically through the bubble functions. Modeling of the unresolved scales corrects the lack of the stability of the standard element-free Galerkin formulation and the resulting stabilized formulation possesses superior properties like that of the streamline upwind/Petrov-Galerkin (SUPG) method and the Galerkin/least-squares (GLS) method. The method allows equal order basis for pressure and velocity because it violates the celebrated Babuska-Brezzi condition. A significant feature of the present method is that the structure of the stabilization tensor $$\tau$$ appears naturally via the solution of the fine-scale problem. Numerical results for example problems confirm that this method has some excellent properties, such as better stability and accuracy.

##### MSC:
 76D07 Stokes and related (Oseen, etc.) flows 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 76M25 Other numerical methods (fluid mechanics) (MSC2010)
Mfree2D
Full Text:
##### References:
 [1] Liu, G.R., Mesh free methods: moving beyond the finite element method, (2003), CRC Press Florida · Zbl 1031.74001 [2] Zhang, X.; Liu, Y., Meshless methods, (2004), Tsinghua University Press Beijing [3] Belytschko, T.; Lu, Y.Y.; Gu, L., Element-free Galerkin methods, Int. J. num. meth. engrg., 37, 229-256, (1994) · Zbl 0796.73077 [4] Belytschko, T.; Krongauz, Y.; Organ, D., Meshless method: an overview and recent developments, Comput. meth. appl. mech. engrg., 139, 3-47, (1996) · Zbl 0891.73075 [5] Belytschko, T.; Krongauz, Y.; Fleming, M., Smoothing and accelerated computations in the element-free Galerkin method, J. comput. appl. math., 74, 111-126, (1996) · Zbl 0862.73058 [6] Liu, W.K.; Li, S.; Belytschko, T., Moving least square kernel methods: (I) methodology and convergence, Comput. meth. appl. mech. engrg., 143, 113-154, (1997) · Zbl 0883.65088 [7] Hughes, T.J.R., Multiscale phenomena: green’s function, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of stabilized methods, Comput. meth. appl. mech. engrg., 127, 387-401, (1995) · Zbl 0866.76044 [8] Hughes, T.J.R.; Feijoo, G.; Mazzei, L.; Quincy, J.B., The variational multiscale method: a paradigm for computational mechanics, Comput. meth. appl. mech. engrg., 166, 3-24, (1998) · Zbl 1017.65525 [9] Hauke, G.; Garcia-Olivares, A., Variational subgrid scale formulations for the advection – diffusion – reaction equation, Comput. meth. appl. mech. engrg., 190, 6847-6865, (2001) · Zbl 0996.76074 [10] Masud, A.; Khurram, R.A., A multiscale/stabilized finite element method for the advection – diffusion equation, Comput. meth. appl. mech. engrg., 193, 1997-2018, (2004) · Zbl 1067.76570 [11] Masud, A.; Khurram, R.A., A multiscale finite element method for the incompressible navier – stokes equations, Comput. meth. appl. mech. engrg., 195, 1750-1777, (2006) · Zbl 1178.76233 [12] Franca, L.P.; Nesliturk, A.; Stynes, M., On the stability of residual-free bubbles for convection – diffusion problems and their approximation by a two-level finite element method, Comput. meth. appl. mech. engrg., 166, 35-49, (1998) · Zbl 0934.65127 [13] Franca, L.P.; Nesliturk, A., On a two-level finite element method for the incompressible navier – stokes equations, Int. J. numer. meth. engrg., 52, 433-453, (2001) · Zbl 1002.76066 [14] A. Nesliturk, Approximating the incompressible Navier-Stokes equations using a two level finite element method, Ph.D. Thesis, Univ. Of Colorado, Denver, 1999. [15] Liu, X.H.; Li, S.F., A variational multiscale stabilized finite element method for the Stokes flow problem, Finite element anal. design, 42, 580-591, (2006) [16] Yeon, Jeoung-Heum; Youn, Sung-Kie, Meshfree analysis of softening elastoplastic solids using variational multiscale method, Int. J. solid struct., 42, 4030-4057, (2005) · Zbl 1120.74867 [17] Lu, Y.Y.; Belytschko, T.; Gu, L., A new implementation of the element-free Galerkin method, Comput. meth. appl. mech. engrg., 113, 397-414, (1994) · Zbl 0847.73064 [18] Chu, Y.A.; Moran, B., A computational model for nucleation of the solid-solid phase transformations, Modell. simul. mater. sci. engrg., 3, 455-471, (1995) [19] Belytschko, T.; Organ, D.; Krongauz, Y., A coupled finite element – element-free Galerkin method, Comput. mech., 17, 186-195, (1995) · Zbl 0840.73058 [20] Huerta, Antonio; Vidal, Yolanda; Villon, Pierre, Pseudo-divergence-free element free Galerkin method for incompressible fluid flow, Comput. meth. appl. mech. engrg., 193, 1119-1136, (2004) · Zbl 1060.76626
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.