Frontiers of Mathematics in China >
Two-grid stabilized mixed finite element method for fully discrete reaction-diffusion equations
Received date: 14 Feb 2015
Accepted date: 08 Sep 2016
Published date: 20 Feb 2017
Copyright
Two-grid mixed finite element method is proposed based on backward Euler schemes for the unsteady reaction-diffusion equations. The scheme combines with the stabilized mixed finite element scheme by using the lowest equal-order pairs for the velocity and pressure. The space two-grid method is also used to reduce the time consuming. The benefits of this approach are to avoid the higher derivative, but to have more favorable stability, and to get the numerical solution of the two unknown variables simultaneously. Stability analysis and error estimates are given in this work. Finally, the theoretical results are verified by the numerical examples.
Sufang ZHANG , Kaitai LI , Hongen JIA . Two-grid stabilized mixed finite element method for fully discrete reaction-diffusion equations[J]. Frontiers of Mathematics in China, 2017 , 12(2) : 481 -492 . DOI: 10.1007/s11464-016-0604-1
1 |
Bochev P B, Dohrmann C R, Gunzburger M D. Stabilized of low-order mixed finite element for the stokes equations. Siam J Numer Anal, 2006, 44(1): 82–101
|
2 |
Bochev P B, Gunzburger M D. An absolutely stable pressure-Poisson stabilized finite element method for Stokes equations. SIAM J Numer Anal, 2004, 42: 1189–1207
|
3 |
Brefort B, Ghidaglia J M, Temam R. Attractor for the penalty Navier-Stokes equations. SIAM J Math Anal, 1988, 19: 1–21
|
4 |
Chen L, Chen Y. Two-grid methods for nonlinear reaction-diffusion equations by expand mixed finite element methods. J Sci Comput, 2011, 49: 383–401
|
5 |
Chen Y, Huang Y, Yu D. A two-grid method for expand mixed finite-element solution of semilinear reaction-diffusion equations. Int J Numer Methods Eng, 2003, 57: 139–209
|
6 |
Chen Y, Li L. Lperror estimates of two-grid schemes of expand mixed finite element methods. Appl Math Comput, 2009, 209: 197–205
|
7 |
Chen Y, Liu H, Liu S. Analysis of two-grid methods for reaction-diffusion equations by expand mixed finite element methods. Int J Numer Methods Eng, 2007, 69: 408–422
|
8 |
Chen Z. Finite Element Methods and Their Applications. Heidelberg: Springer-Verlag, 2005
|
9 |
Dawson C, Kirby R. Solution of parabolic equations by backward euler mixed finite element methods on a dynamically changing mesh. SIAM J Numer Anal, 1999, 37: 423–442
|
10 |
de Freitas J A T. Mixed finite element formulation for the solution of parabolic problem. Comput Methods Appl Mech Engrg, 2002, 191: 3425–3457
|
11 |
Gilbarg D, Trudinger N S. Elliptic Partial Differential Equations of Second Order. Berlin: Springer, 2001
|
12 |
He Y, Li K. Two-grid stabilized finite element methods for the stead Navier-Stokes problem. Computing, 2005, 74: 337–351
|
13 |
Hughes T, Franca L, Balestra M. A new finite element formulation for computational uid dynamics: V. Circumventing the Babuska-Brezzi condition: a stable Petrov-Galerkin formulation of the Stokes problem accommodating equal-order interpolations. Comput Methods Appl Mech Engrg, 1986, 59: 85–89
|
14 |
Huyakorn P S, Pinder G F. Computational Methods in Subsurface Flow. New York: Academic Press, 1983
|
15 |
Li J, He Y. A stabilized finite element method based on local polynomial pressure projection for the stationary Navier-Stokes equations. Appl Numer Math, 2008, 58(10): 1503–1514
|
16 |
Li J, He Y. A stabilzied finite element method based on two local Guass integrations for the Stokes equations. J Comput Appl Math (to appear)
|
17 |
Li J, Mei L, He Y. A pressure-Poissuon stabilized finite element method for the non-stationary Stokes equations to circumvent the inf-sup condition. Appl Math Comput, 2006, 1: 24–35
|
18 |
Masud A, Hughes T J R. A stabilized finite element method for Darcy ow. Comput Methods Appl Mech Engrg, 2002, 191: 4341–4370
|
19 |
Murray J. Mathematical Biology. 2nd ed. New York: Springer, 1993
|
20 |
Nakshatrala K B, Turner D Z, Hjelmstad K D, Masud A. A stabilized mixed finite element method for Darcy ow based on a multiscale decomposition of the solution. Comput Methods Appl Mech Engrg, 2006, 195: 4036–4049
|
21 |
Rui H X. Symmetric mixed covolume methods for parabolic problem. Numer Methods Partial Differential Equations, 2002, 18: 561–583
|
22 |
Shang Y Q. New stabilized finite element method for time-dependent incompressible ow problems. Int J Numer Methods Fluids, 2009, www.interscience.wiley.com, DOI:10.1002/d.2010
|
23 |
Smith B, Bjorstad P, Grropp W. Domain Decomposition, Parallel Multilevel Method for Elliptic Partial Differential Equations. Cambridge: Cambridge Univ Press, 1996
|
24 |
Xu J. A new class of iterative methods for nonselfadjoint or indefinite elliptic problems. SIAM J Numer Anal, 1992, 29: 303–319
|
25 |
Zhang L, Chen Z. A stabilized mixed finite element method for single-phase compressible ow. J Appl Math, 2011, 2011(2): 430–441
|
/
〈 | 〉 |