zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
A stabilized mixed finite element method for single-phase compressible flow. (English) Zbl 1204.76022
Summary: We present and study a stabilized mixed finite element method for single-phase compressible flow through porous media. This method is based on a pressure projection stabilization method for multiple-dimensional incompressible flow problems by using the lowest equal-order pair for velocity and pressure (i.e., the $P_1 - P_1$ pair). An optimal error estimate in divergence norm for the velocity and suboptimal error estimates in the $L^2$-norm for both velocity and pressure are obtained. Numerical results are given in support of the developed theory.

MSC:
76M10Finite element methods (fluid mechanics)
WorldCat.org
Full Text: DOI EuDML
References:
[1] Z. Chen, Finite Element Methods and Their Applications, Scientific Computation, Springer, Berlin, Germany, 2005.
[2] V. Girault and P. A. Raviart, Finite Element Methods for Navier-Stokes Equations, vol. 5 of Springer Series in Computational Mathematics, Springer, Berlin, Germany, 1986. · Zbl 0585.65077
[3] D. N. Arnold, F. Brezzi, and M. Fortin, “A stable finite element for the Stokes equations,” Calcolo, vol. 21, no. 4, pp. 337-344, 1984. · Zbl 0593.76039 · doi:10.1007/BF02576171
[4] J.-C. Nedelec, “Mixed finite elements in R3,” Numerische Mathematik, vol. 35, no. 3, pp. 315-341, 1980. · Zbl 0419.65069 · doi:10.1007/BF01396415 · eudml:186293
[5] R. A. Raviart and J. M. Thomas, “A mixed FInite element method for 2nd order elliptic problems,” in Mathematical Aspects of Finite Element Methods, I. Galligani and E. Magenes, Eds., vol. 606 of Lecture Notes in Mathematics, pp. 292-315, Springer, New York, NY, USA, 1997.
[6] F. Brezzi, J. Douglas, Jr., and L. D. Marini, “Two families of mixed finite elements for second order elliptic problems,” Numerische Mathematik, vol. 47, no. 2, pp. 217-235, 1985. · Zbl 0599.65072 · doi:10.1007/BF01389710 · eudml:133032
[7] F. Brezzi, J. Douglas, Jr., R. Durán, and M. Fortin, “Mixed finite elements for second order elliptic problems in three variables,” Numerische Mathematik, vol. 51, no. 2, pp. 237-250, 1987. · Zbl 0631.65107 · doi:10.1007/BF01396752 · eudml:133194
[8] Z. Chen and J. Douglas, Jr., “Prismatic mixed finite elements for second order elliptic problems,” Calcolo, vol. 26, no. 2-4, pp. 135-148, 1989. · Zbl 0711.65089 · doi:10.1007/BF02575725
[9] F. Brezzi, M. Fortin, and L. D. Marini, “Mixed finite element methods with continuous stresses,” Mathematical Models & Methods in Applied Sciences, vol. 3, no. 2, pp. 275-287, 1993. · Zbl 0774.73066 · doi:10.1142/S0218202593000151
[10] J. Li and Y. He, “A new stabilized finite element method based on local Gauss integration techniques for the Stokes equations,” Journal of Computational and Applied Mathematics, vol. 214, pp. 58-65, 2008. · Zbl 1132.35436
[11] R. Becker and M. Braack, “A finite element pressure gradient stabilization for the Stokes equations based on local projections,” Calcolo, vol. 38, no. 4, pp. 173-199, 2001. · Zbl 1008.76036 · doi:10.1007/s10092-001-8180-4
[12] P. Bochev, C. R. Dohrmann, and M. D. Gunzburger, “Stabilization of low-order mixed finite elements for the Stokes equations,” SIAM Journal on Numerical Analysis, vol. 44, no. 1, pp. 82-101, 2006. · Zbl 1145.76015 · doi:10.1137/S0036142905444482
[13] F. Brezzi and M. Fortin, “A minimal stabilisation procedure for mixed finite element methods,” Numerische Mathematik, vol. 89, no. 3, pp. 457-491, 2001. · Zbl 1009.65067 · doi:10.1007/s002110100258
[14] C. R. Dohrmann and P. Bochev, “A stabilized finite element method for the Stokes problem based on polynomial pressure projections,” International Journal for Numerical Methods in Fluids, vol. 46, no. 2, pp. 183-201, 2004. · Zbl 1060.76569 · doi:10.1002/fld.752
[15] J. Li, Y. He, and Z. Chen, “A new stabilized finite element method for the transient Navier-Stokes equations,” Computer Methods in Applied Mechanics and Engineering, vol. 197, no. 1-4, pp. 22-35, 2007. · Zbl 1169.76392 · doi:10.1016/j.cma.2007.06.029
[16] D. Silvester, “Stabilized mixed finite element methods,” Numerical Analysis Report 262, 1995.
[17] R. Falk, “An analysis of the penalty method and extrapolation for the stationary Stokes equations,” in Advances in Computer Methods for Partial Differential Equations, R. Vichnevetsky, Ed., pp. 66-69, AICA, 1975.
[18] T. J. R. Hughes, W. Liu, and A. Brooks, “Finite element analysis of incompressible viscous flows by the penalty function formulation,” Journal of Computational Physics, vol. 30, no. 1, pp. 1-60, 1979. · Zbl 0412.76023 · doi:10.1016/0021-9991(79)90086-X
[19] A. Masud and T. J. R. Hughes, “A stabilized mixed finite element method for Darcy flow,” Computer Methods in Applied Mechanics and Engineering, vol. 191, no. 39-40, pp. 4341-4370, 2002. · Zbl 1015.76047 · doi:10.1016/S0045-7825(02)00371-7
[20] Z. Chen, Z. Wang, and J. Li, “Analysis of the pressure projectionstabilization method for second-order elliptic problems,” to appear.
[21] J. Li, Y. He, and Z. Chen, “Performance of several stabilized finite element methods for the Stokes equations based on the lowest equal-order pairs,” Computing. Archives for Scientific Computing, vol. 86, no. 1, pp. 37-51, 2009. · Zbl 1176.65136 · doi:10.1007/s00607-009-0064-5
[22] Z. Chen, G. Huan, and Y. Ma, Computational Methods for Multiphase Flows in Porous Media, SIAM, Philadelphia, Pa, USA, 2006. · Zbl 1092.76001
[23] P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, The Netherlands, 1978. · Zbl 0383.65058