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Two-grid methods for finite volume element approximations of nonlinear parabolic equations. (English) Zbl 1169.65094
The authors prove an error estimate for a finite-volume method for a reaction-diffusion equation on convex polygonal domains in the plane. The reaction term is nonlinear. At each time step the discrete nonlinear equation is solved on a coarse grid, and the result is used as an initial approximation for a Newton iteration on a fine grid.

MSC:
65M55Multigrid methods; domain decomposition (IVP of PDE)
65M15Error bounds (IVP of PDE)
35K57Reaction-diffusion equations
65M06Finite difference methods (IVP of PDE)
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References:
[1] Thomée, V.: Galerkin finite element methods for parabolic problems, (1997) · Zbl 0884.65097
[2] Cai, Z.: On the finite volume element methods, Numer. math. 58, 713-735 (1991) · Zbl 0731.65093
[3] Cai, Z.; Mccormick, S.: On the accuracy of the finite volume element method for diffusion equations on composite grids, SIAM J. Numer. anal. 27, 636-655 (1990) · Zbl 0707.65073 · doi:10.1137/0727039
[4] Ewing, R. E.; Lazarov, R. D.; Lin, Y. P.: Finite volume element approximations of nonlocal reactive flows in porous media, Numer. methods partial differential equations 16, 285-311 (2000) · Zbl 0961.76050 · doi:10.1002/(SICI)1098-2426(200005)16:3<285::AID-NUM2>3.0.CO;2-3
[5] Ewing, R. E.; Lin, T.; Lin, Y. P.: On the accuracy of the finite volume element method based on piecewise linear polynomials, SIAM J. Numer. anal. 39, No. 6, 1865-1888 (2002) · Zbl 1036.65084 · doi:10.1137/S0036142900368873
[6] Bank, R. E.; Rose, D. J.: Some error estimates for the box method, SIAM J. Numer. anal. 24, 777-787 (1987) · Zbl 0634.65105 · doi:10.1137/0724050
[7] Hackbusch, W.: On first and second order box schemes, Computing 41, 277-296 (1989) · Zbl 0649.65052 · doi:10.1007/BF02241218
[8] Mishev, I. D.: Finite volume methods on Voronoi meshes, Numer. methods partial differential equations 14, 193-212 (1998) · Zbl 0903.65083 · doi:10.1002/(SICI)1098-2426(199803)14:2<193::AID-NUM4>3.0.CO;2-J
[9] Li, R.; Chen, Z.; Wu, W.: Generalized difference methods for differential equations numerical analysis of finite volume methods, (2000) · Zbl 0940.65125
[10] Chatzipantelidis, P.; Lazarov, R. D.; Thomée, V.: Error estimate for a finite volume element method for parabolic equations in convex polygonal domains, Numer. methods partial differential equations 20, 650-674 (2004) · Zbl 1067.65092 · doi:10.1002/num.20006
[11] Chatzipantelidis, P.: Finite volume methods for elliptic PDE’s: A new approach, M2AN math. Model. numer. Anal. 36, 307-324 (2002) · Zbl 1041.65087 · doi:10.1051/m2an:2002014 · numdam:M2AN_2002__36_2_307_0
[12] Xu, J.: A novel two-grid method for semilinear elliptic equations, SIAM J. Sci. comput. 15, 231-237 (1994) · Zbl 0795.65077 · doi:10.1137/0915016
[13] Xu, J.: Two-grid discretization techniques for linear and nonlinear pdes, SIAM J. Numer. anal. 33, 1759-1777 (1996) · Zbl 0860.65119 · doi:10.1137/S0036142992232949
[14] Dawson, C. N.; Wheeler, M. F.: Two-grid methods for mixed finite element approximations of nonlinear parabolic equations, Contemp. math. 180, 191-203 (1994) · Zbl 0817.65080
[15] Dawson, C. N.; Wheeler, M. F.; Woodward, C. S.: A two-grid finite difference scheme for nonlinear parabolic equations, SIAM J. Numer. anal. 35, 435-452 (1998) · Zbl 0927.65107 · doi:10.1137/S0036142995293493
[16] Wu, L.; Allen, M. B.: A two-grid method for mixed finite-element solutions of reaction-diffusion equations, Numer. methods partial differential equations 15, 589-604 (1999) · Zbl 0942.65106 · doi:10.1002/(SICI)1098-2426(199909)15:5<589::AID-NUM6>3.0.CO;2-W
[17] Chen, Y.; Huang, Y.; Yu, D.: A two-grid method for expanded mixed finite-element solution of semilinear reaction-diffusion equations, Internat. J. Numer. methods engrg 57, 139-209 (2003) · Zbl 1062.65104 · doi:10.1002/nme.668
[18] Bi, C.; Ginting, V.: Two-grid finite volume element method for linear and nonlinear elliptic problems, Numer. math. 108, 177-198 (2007) · Zbl 1134.65077 · doi:10.1007/s00211-007-0115-9
[19] Kuznetsov, Yu.A.: New algorithm for approximate realization of implicit difference scheme, Sov. J. Numer. anal. Math. modeling. 3, 95-114 (1988) · Zbl 0825.65066
[20] Yu.A. Kuznestsov, Domain decomposition methods for unsteady convection-diffusion problems, In: Proc. 9-th Int. Conf. Computing Methods in Appl. Sci. Eng., 1990. pp. 211--227 · Zbl 0744.65063
[21] Rannacher, R.; Zhou, G.: Analysis of a domain-splitting method for non-stationary convection-diffusion problems, East-west J. Numer. math. 2, 151-172 (1994) · Zbl 0836.65100
[22] Brenner, S. C.; Scott, L. R.: The mathematical theory of finite element methods, (1994) · Zbl 0804.65101
[23] Russell, T. F.: Time stepping along characteristics with incomplete iteration for a Galerkin approximation of miscible displacement in porous media, SIAM J. Numer. anal. 22, No. 5, 970-1013 (1985) · Zbl 0594.76087 · doi:10.1137/0722059
[24] Chou, S. H.; Li, Q.: Error estimates in L2, H1 and L$\infty $in covolume methods for elliptic and parabolic problems: A unified approach, Math. comp. 69, 103-120 (2000) · Zbl 0936.65127 · doi:10.1090/S0025-5718-99-01192-8