A two-grid expanded mixed element method for nonlinear non-Fickian flow model in porous media

References

• W. Allegretto, Y. Lin, and A. Zhou, A box scheme for coupled systems resulting from microsensor thermistor problems, Dyn. Discrete Contin. Impuls. Syst. 5 (1995), pp. 209–223.
   Google Scholar
• A. Arikoglu and I. Ozkol, Solution of boundary value problems for integro-differential equations by using differential transform method, Appl. Math. Comput. 168 (2005), pp. 1145–1158. doi: 10.1016/j.amc.2004.10.009
   Web of Science ®Google Scholar
• C. Bi and V. Ginting, Two-grid finite volume element for linear and nonlinear elliptic problems, Numer. Math. 108 (2007), pp. 177–198. doi: 10.1007/s00211-007-0115-9
   Web of Science ®Google Scholar
• J.R. Cannon and Y. Lin, Non-classical H1 projection and Galerkin methods for nonlinear parabolic integro-differential equations, Calcolo 25 (1988), pp. 187–201. doi: 10.1007/BF02575943
   Google Scholar
• J.R. Cannon and Y. Lin, A priori L2 error estimates for finite element methods for nonlinear diffusion equations with memory, SIAM. J. Numer. Anal. 27 (1990), pp. 595–607. doi: 10.1137/0727036
   Web of Science ®Google Scholar
• Z. Chen, Expanded mixed element methods for linear second-order elliptic problems(I), RAIRO Model. Math. Anal. Numer. 32 (1998), pp. 479–499.
   Web of Science ®Google Scholar
• Z. Chen, Expanded mixed element methods for quasilinear second-order elliptic problems(II), RAIRO Model. Math. Anal. Numer. 32 (1998), pp. 501–520.
   Web of Science ®Google Scholar
• Y. Chen, Y. Huang, and D. Yu, A two-grid method for expanded mixed finite-element solution of semilinear reaction-diffusion equations, Int. J. Numer. Methods Eng. 57 (2003), pp. 193–209. doi: 10.1002/nme.668
   Web of Science ®Google Scholar
• C.N. Dawson and M.F. Wheeler, Two-grid methods for mixed finite element approximations of nonlinear parabolic equations, Contemp. Math. 180 (1994), pp. 191–203. doi: 10.1090/conm/180/01971
   Google Scholar
• E. Deeba, S.A. Khuri, and S. Xie, An algorithm for solving a nonlinear integro-differential equation, Appl. Math. Comput. 115 (2000), pp. 123–131. doi: 10.1016/S0096-3003(99)00124-1
   Web of Science ®Google Scholar
• J. DouglasJr. and J. Wang, Superconvergence of mixed finite element methods on rectangular domains, CALCOLO 26 (1989), pp. 121–133. doi: 10.1007/BF02575724
   Google Scholar
• R.E. Ewing, R.D. Lazarov, and J. Wang, Superconvergence of the velocity along the Gauss lines in mixed finite element methods, SIAM J. Numer. Anal. 28 (1991), pp. 1015–1029. doi: 10.1137/0728054
   Web of Science ®Google Scholar
• R.E. Ewing, Y. Lin, T. Sun, J. Wang, and S. Zhang, Sharp L2-error estimates and superconvergence of mixed finite element methods for non-Fichian flows in porous media, SIAM J. Numer. Anal. 40 (2002), pp. 1538–1500. doi: 10.1137/S0036142900378406
   Web of Science ®Google Scholar
• M. Gurtin and A. Pipkin, A general theory of heat conduction with finite wave speeds, Arch. Ration. Mech. Anal. 31 (1968), pp. 113–126. doi: 10.1007/BF00281373
   Web of Science ®Google Scholar
• Y. He, Two-level method based on finite element and Crank-Nicolson extrapolation for the time-dependent Navier–Stokes equations, SIAM J. Numer. Anal. 41 (2003), pp. 1263–1285. doi: 10.1137/S0036142901385659
   Web of Science ®Google Scholar
• Y. He, A two-level finite element method for the nonstationary Navier–Stokes equations I: Spatial discretization, J. Comput. Math. 22 (2004), pp. 21–32.
   Web of Science ®Google Scholar
• Y. He and K. Li, Two-level stabilized finite element methods for the Steady Navier–Stokes problem, Computing 74 (2005), pp. 367–380. doi: 10.1007/s00607-004-0118-7
   Web of Science ®Google Scholar
• Y. He and K.M. Liu, A multi-level finite element method in spacetime for the Navier–Stokes equations, Numer. Meth. PDEs 21 (2005), pp. 1052–1078. doi: 10.1002/num.20077
   Web of Science ®Google Scholar
• Y. He and A. Wang, A simplified two-level method for the steady Navier–Stokes equations, Comput. Methods Appl. Mech. Eng. 197 (2008), pp. 1568–1576. doi: 10.1016/j.cma.2007.11.032
   Web of Science ®Google Scholar
• Y. He, H. Miao, and C. Ren, A two-level finite element method for the nonstationary Navier–Stokes equations II: Time discretization, J. Comput. Math. 22 (2004), pp. 33–54.
   Web of Science ®Google Scholar
• W. Liu, H. Rui, and H. Guo, A two-grid method with expanded mixed element for nonlinear reaction–diffusion equations, Acta Math. Appl. Sin. E 27 (2011), pp. 495–502. doi: 10.1007/s10255-011-0086-6
   Web of Science ®Google Scholar
• W. Liu, H. Rui, and F. Hu, A two-grid algorithm for expanded mixed finite element approximations of semi-linear elliptic equations, Comput. Math. Appl. 66 (2013), pp. 392–402. doi: 10.1016/j.camwa.2013.05.016
   Web of Science ®Google Scholar
• F. Milner, Mixed finite element methods for quasilinear second order elliptic problems, Math. Comput. 44 (1985), pp. 303–320. doi: 10.1090/S0025-5718-1985-0777266-1
   Google Scholar
• J.C. Nedelec, Mixed finite elements in R3, Numer. Math. 35 (1990), pp. 315–341. doi: 10.1007/BF01396415
   Google Scholar
• V.V. Shelukin, A nonlocal in time model for radionuclides propagation in stokes fluids dynamics of fluids with free boundaries, Siberian Russ. Acad. Sci. Inst. Hydrodyn. 107 (1993), pp. 180–193.
   Google Scholar
• C. Wang, Z. Huang, and L. Li, Two-grid covolume schemes for elliptic problems, Commun. Numer. Methods Eng. 22 (2006), pp. 397–409. doi: 10.1002/cnm.822
   Web of Science ®Google Scholar
• J. Xu, Two-grid discretization techniques for linear and nonlinear PDEs, SIAM J. Numer. Anal. 33 (1996), pp. 1759–1777. doi: 10.1137/S0036142992232949
   Web of Science ®Google Scholar