zbMATH — the first resource for mathematics

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.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
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 new mixed finite element method based on the Crank-Nicolson scheme for the parabolic problems. (English) Zbl 1252.65170
Summary: We present a new mixed finite element method based on the less regularity of flux (velocity) for the parabolic problems in practice. Based on this new formulation, we give its corresponding stable conforming finite element approximation for the P 0 2 -P 1 pair by using Crank-Nicolson time-discretization scheme. Moreover, we derive optimal error estimates in H 1 -norm and L 2 -norm for the approximation of pressure p and in L 2 -norm for the approximation of velocity u. Finally, we give some numerical experiments to verify the efficiency and the theoretical results of the new mixed method.
65M60Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (IVP of PDE)
[1]Nedelec, J. C.: Mixed finite elements in R3, Numer. math. 35, 315-341 (1980) · Zbl 0419.65069 · doi:10.1007/BF01396415
[2]Raviart, R. A.; Thomas, J. M.: A mixed finite element method for 2nd order elliptic problems, Lecture notes in mathematics 606 (1977) · Zbl 0362.65089
[3]Jr., J. Douglas; Roberts, J. E.: Global estimates for mixed methods for second order elliptic equations, Math. comput. 44, 39-52 (1985) · Zbl 0624.65109 · doi:10.2307/2007791
[4]Chen, H. S.; Ewing, R.; Lazarav, R.: Superconvergence of mixed finite element methods for parabolic problems with nonsmooth initial data, Numer. math. 78, 495-521 (1998) · Zbl 0894.65044 · doi:10.1007/s002110050323
[5]Brezzi, F.; Jr., J. Douglas; Marini, L. D.: Two families of mixed finite elements for second order elliptic problems, Numer. math. 47, 217-235 (1985) · Zbl 0599.65072 · doi:10.1007/BF01389710
[6]Brezzi, F.; Jr., J. Douglas; Duran, R.; Fortin, M.: Mixed finite elements for second order elliptic problems in three variables, Numer. math. 51, 237-250 (1987) · Zbl 0631.65107 · doi:10.1007/BF01396752
[7]Brezzi, F.; Fortin, M.; Marini, L. D.: Mixed finite element methods with continuous stresses, Math. models methods appl. Sci. 3, 275-287 (1993) · Zbl 0774.73066 · doi:10.1142/S0218202593000151
[8]Chen, Z.; Jr., J. Douglas: Prismatic mixed finite elements for second order elliptic problems, Calcolo 26, 135-148 (1989) · Zbl 0711.65089 · doi:10.1007/BF02575725
[9]Shi, F.; Yu, J. P.; Li, K. T.: A new stabilized mixed finite-element method for Poisson equation based on two local Gauss integrations for linear element pair, Int. J. Comput. math. 88, 2293-2305 (2011)
[10]Z.F. Weng, X.L. Feng, D.M. Liu, A fully discrete stabilized mixed finite element method for the parabolic problems, preprint.
[11]Johnson, C.: Numerical solution of partial differential equations by the finite element method, (1987) · Zbl 0628.65098
[12]Crank, J.; Nicolson, P.: A practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type, Math. proc. Camb. philos. Soc. 43, 50-67 (1947) · Zbl 0029.05901
[13]He, Y. N.; Sun, W. W.: Stabilized finite element method based on the Crank – Nicolson extrapolation scheme for the time-dependent Navier – Stokes equations, Math. comput. 76, 115-136 (2007) · Zbl 1129.35004 · doi:10.1090/S0025-5718-06-01886-2
[14]He, Y. N.; Sun, W. W.: Stability and convergence of the Crank – Nicolson/Adams – bashforth scheme for the time-dependent Navier – Stokes equations, SIAM J. Numer. anal. 45, 837-869 (2007) · Zbl 1145.35318 · doi:10.1137/050639910
[15]He, Y. N.: Two-level method based on finite element and Crank – Nicolson extrapolation for the time-dependent Navier – Stokes equations, SIAM J. Numer. anal. 41, 1263-1285 (2003) · Zbl 1130.76365 · doi:10.1137/S0036142901385659
[16]Heywood, J. G.; Rannacher, R.: Finite-element approximations of the nonstationary Navier – Stokes problem. Part IV: Error estimates for second-order time discretization, SIAM J. Numer. anal. 27, 353-384 (1990) · Zbl 0694.76014 · doi:10.1137/0727022
[17]Feng, X. L.; He, Y. N.; Liu, D. M.: Convergence analysis of an implicit fractional-step method for the incompressible Navier – Stokes equations, Appl. math. Model. 35, 5856-5871 (2011) · Zbl 1228.76040 · doi:10.1016/j.apm.2011.05.042
[18]Feng, X. L.; He, Y. N.: The convergence of a new parallel algorithm for the Navier – Stokes equations, Nonlinear anal. Real world appl. 10, 23-41 (2009) · Zbl 1154.65355 · doi:10.1016/j.nonrwa.2007.08.011
[19]Tone, F.: Error analysis for a second scheme for the Navier – Stokes equations, Appl. numer. Math. 50, 93-119 (2004) · Zbl 1093.76046 · doi:10.1016/j.apnum.2003.12.003
[20]Johnston, H.; Liu, J. G.: Accurate, stable and efficient Navier – Stokes slovers based on explicit treatment of the pressure term, J. comput. Phys. 199, 221-259 (2004) · Zbl 1127.76343 · doi:10.1016/j.jcp.2004.02.00
[21]Ciarlet, P. G.: The finite element method for elliptic problems, (1978)
[22]Brezzi, F.; Fortin, M.: Mixed and hybrid finite element methods, (1991) · Zbl 0788.73002
[23]FreeFem++, version 2.19.1, lt;http://www.freefem.org/gt;.