Two-grid stabilized mixed finite element method for fully discrete reaction-diffusion equations
Sufang ZHANG, Kaitai LI, Hongen JIA
Two-grid stabilized mixed finite element method for fully discrete reaction-diffusion equations
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.
Reaction-diffusion equations / stabilized mixed finite element / two-grid / full-discrete schemes
[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
CrossRef
Google scholar
|
[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
CrossRef
Google scholar
|
[3] |
Brefort B, Ghidaglia J M, Temam R. Attractor for the penalty Navier-Stokes equations. SIAM J Math Anal, 1988, 19: 1–21
CrossRef
Google scholar
|
[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
CrossRef
Google scholar
|
[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
CrossRef
Google scholar
|
[6] |
Chen Y, Li L. Lperror estimates of two-grid schemes of expand mixed finite element methods. Appl Math Comput, 2009, 209: 197–205
CrossRef
Google scholar
|
[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
CrossRef
Google scholar
|
[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
CrossRef
Google scholar
|
[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
CrossRef
Google scholar
|
[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
CrossRef
Google scholar
|
[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
CrossRef
Google scholar
|
[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
CrossRef
Google scholar
|
[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
CrossRef
Google scholar
|
[18] |
Masud A, Hughes T J R. A stabilized finite element method for Darcy ow. Comput Methods Appl Mech Engrg, 2002, 191: 4341–4370
CrossRef
Google scholar
|
[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
CrossRef
Google scholar
|
[21] |
Rui H X. Symmetric mixed covolume methods for parabolic problem. Numer Methods Partial Differential Equations, 2002, 18: 561–583
CrossRef
Google scholar
|
[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
CrossRef
Google scholar
|
[25] |
Zhang L, Chen Z. A stabilized mixed finite element method for single-phase compressible ow. J Appl Math, 2011, 2011(2): 430–441
|
/
〈 | 〉 |