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The semidiscrete finite volume element method for nonlinear convection-diffusion problem. (English) Zbl 1221.65234

A nonlinear convection-diffusion problem in 2 is solved numerically with a semi-discrete finite volume method. The basic idea is to approximate the discrete fluxes of the partial differential equation using a finite element procedure based on control volumes. Results on the stability of the semi-discrete method as well as the existence and uniqueness of the solution provided by the scheme are provided.

In the second part of the paper, a two-grid formulation of the (semi-discrete) finite volume method is introduced to solve the nonlinear parabolic problem. In this approach, two regular triangulations of the domain with mess size H and hH are considered. The nonlinear problem is solved directly on the coarse grid and then the provided solution is used to construct the fine grid solution. The original problem becomes linear in this second stage of the procedure, and thus is much simpler to solve. L 2 -norm and H 1 -norm error estimates are also provided. A numerical example is included to illustrate the theoretical analysis.


MSC:
65M08Finite volume methods (IVP of PDE)
65M20Method of lines (IVP of PDE)
35K55Nonlinear parabolic equations
65M12Stability and convergence of numerical methods (IVP of PDE)
65M15Error bounds (IVP of PDE)
65M60Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (IVP of PDE)
65M55Multigrid methods; domain decomposition (IVP of PDE)
References:
[1]Bank, R. E.; Rose, D. J.: Some error estimates for the box method, SIAM J. Numer. anal. 24, No. 4, 777-787 (1987) · Zbl 0634.65105 · doi:10.1137/0724050
[2]Bi, C. J.; Ginting, V.: Two-grid finite volume element method for linear and nonlinear elliptic problems, Numer. math. 108, No. 2, 177-198 (2007) · Zbl 1134.65077 · doi:10.1007/s00211-007-0115-9
[3]Brenner, S. C.; Scott, L. R.: The mathematics theory of finite element methods, (1994)
[4]Cai, Z.; Jones, J. E.; Mccormick, S. F.; Russell, T. F.: Control-volume mixed finite element methods, Comput. geosci. 1, No. 3, 289-315 (1997) · Zbl 0941.76050 · doi:10.1023/A:1011577530905
[5]Cai, Z.; Mandel, J.; Mccormick, S.: The finite volume element method for diffusion equations on general triangulations, SIAM J. Numer. anal. 28, No. 2, 392-402 (1991) · Zbl 0729.65086 · doi:10.1137/0728022
[6]Chen, Y.; Yang, M.; Bi, C. J.: Two-grid methods for finite volume element approximations of nonlinear parabolic equations, J. comput. Applied math. 228, No. 1, 123-132 (2009) · Zbl 1169.65094 · doi:10.1016/j.cam.2008.09.001
[7]Chou, S. H.; Li, Q.: Error estimates in L2, H1 and L in covolume methods for elliptic and parabolic problem: A unified approach, Math. comp. 69, No. 229, 103-120 (2000) · Zbl 0936.65127 · doi:10.1090/S0025-5718-99-01192-8
[8]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
[9]Dawson, C. N.; Wheeler, M. F.; Woodward, C. S.: A two-grid finite difference scheme for nonlinear parabolic equations, SIAM J. Numer. anal. 35, No. 2, 435-452 (1998) · Zbl 0927.65107 · doi:10.1137/S0036142995293493
[10]Dolejsi, V.; Feistauer, M.; Schwab, C.: A finite volume discontinuous Galerkin scheme for nonlinear convection – diffusion problems, Calcolo 39, No. 1, 1-40 (2002) · Zbl 1098.65095 · doi:10.1007/s100920200000
[11]Drblikova, O.; Mikula, K.: Convergence analysis of finite volume scheme for nonlinear tensor anisotropic diffusion in image processing, SIAM J. Numer. anal. 46, No. 1, 37-60 (2007) · Zbl 1233.65063 · doi:10.1137/070685038
[12]Ewing, R.; Iliev, O.; Lazarov, R.: A modified finite volume approximation of second-order elliptic equations with discontinuous coefficients, SIAM J. Sci. comput. 23, No. 4, 1335-1351 (2001) · Zbl 0999.65112 · doi:10.1137/S1064827599353877
[13]Ewing, R. E.; Lin, T.; Lin, Y. P.: On the accuracy of the finite volume element based on piecewise linear polynomials, SIAM J. Numer. anal. 39, No. 6, 1865-1888 (2002) · Zbl 1036.65084 · doi:10.1137/S0036142900368873
[14]Eymard, R.; Gallouet, T.; Herbin, R.: Finite volume methods, Handbook of numerical analysis 7 (2000) · Zbl 0981.65095
[15]Feistauer, M.; Felcaman, J.; Lukacova-Medvidova, M.; Warnecke, G.: Error estimates for a combined finite volume-finite element method for nonlinear convection – diffusion problems, SIAM J. Numer. anal. 36, No. 5, 1528-1548 (1999) · Zbl 0960.65098 · doi:10.1137/S0036142997314695
[16]He, G. L.; He, Y. N.; Feng, X. L.: Finite volume method based on stabilized finite elements for the nonstationary Navier – Stokes problem, Numerical method for pdes 23, No. 5, 1167-1191 (2007) · Zbl 1127.76036 · doi:10.1002/num.20216
[17]Hegarty, A. F.; Miller, J. J. H.; Oriordan, E.: Special meshes for finite difference approximations to an advection – diffusion equation with parabolic layers, J. comput. Physics 117, No. 1, 47-54 (1995) · Zbl 0817.76041 · doi:10.1006/jcph.1995.1043
[18]Hill, A. T.; Suli, E.: Approximation of the global attractor for the incompressible Navier – Stokes equations, IMA J. Numer. anal. 20, 1321-1337 (2000) · Zbl 0982.76022 · doi:10.1093/imanum/20.4.633
[19]Gao, F. Z.: Finite volume element predictor – corrector method for a class of nonlinear parabolic systems, Northeast math. J. 21, No. 3, 305-314 (2005) · Zbl 1092.65082
[20]Gao, F. Z.; Yuan, Y. R.: The characteristic finite volume element method for the nonlinear convection-dominated diffusion problem, Comput. math. Appl. 56, No. 1, 71-81 (2008) · Zbl 1145.65320 · doi:10.1016/j.camwa.2007.11.033
[21]Li, R. H.: Generalized difference methods for a nonlinear Dirichlet problem, SIAM J. Numer. anal. 24, No. 1, 77-88 (1987) · Zbl 0626.65091 · doi:10.1137/0724007
[22]Li, R. H.; Chen, Z.; Wu, W.: Generalized difference methods for differential equations numerical analysis of finite volume methods, (2000)
[23]Layton, W.; Lenferink, W.: Two-level Picard and modified Picard methods for the Navier – Stokes equations, Appl. math. Comput. 69, No. 1, 263-274 (1995) · Zbl 0828.76017 · doi:10.1016/0096-3003(94)00134-P
[24]Mikula, K.; Ramarosy, N.: Semi-implicit finite volume scheme for solving nonlinear diffusion equations in image processing, Numer. math. 89, No. 3, 561-590 (2001) · Zbl 1013.65094 · doi:10.1007/s002110100264
[25]Morton, K. W.: Numerical solution of convection – diffusion problems, (1996)
[26]Marion, M.; X, J.: Error estimates on a new nonlinear Galerkin method based on two-grid finite elements, SIAM J. Numer. anal. 32, 1170-1184 (1995) · Zbl 0853.65092 · doi:10.1137/0732054
[27]Roos, H. G.; Stynes, M.; Tobiska, L.: Numerical methods for singularly perturbed differential equations, convection – diffusion and flow problems, (1996)
[28]Sinha, R. K.; Ewing, R. E.; Lazarov, R. D.: Some new error estimates of a semidiscrete finite volume element method for a parabolic integro-differential equation with nonsmooth initial data, SIAM J. Numer. anal. 43, No. 6, 2320-2343 (2006) · Zbl 1106.65121 · doi:10.1137/040612099
[29]Thomée, V.: Galerkin finite element method for parabolic problems, (1997)
[30]Varah, J. M.: Stability restrictions on second order, three level finite difference schemes for parabolic equations, SIAM J. Numer. anal. 17, No. 2, 300-309 (1980) · Zbl 0426.65048 · doi:10.1137/0717025
[31]Wu, H. J.; Li, R. H.: Error estimates for fnite volume element methods for general second order elliptic problem, Numer. meth. Pdes 19, 693-708 (2003) · Zbl 1040.65091 · doi:10.1002/num.10068
[32]Xu, J. C.: A novel two-grid method for semi-linear elliptic equations, SIAM J. Sci. comput. 15, No. 1, 231-237 (1994) · Zbl 0795.65077 · doi:10.1137/0915016
[33]Xu, J. C.: Two-grid discretization techniques for linear and nonlinear pdes, SIAM J. Numer. anal. 33, No. 5, 1759-1777 (1996) · Zbl 0860.65119 · doi:10.1137/S0036142992232949
[34]Xu, J. C.; Zou, Q.: Analysis of linear and quadratic simplicial finite volume methods for elliptic equations, Numer. math. 111, 469-492 (2009) · Zbl 1169.65110 · doi:10.1007/s00211-008-0189-z
[35]T.Zhang, H.Zhong, J.Zhao, A full discrete two-grid finite-volume method for a nonlinear parabolic problem, Int. J. Comput. Math., to appear.*** · Zbl 1221.65235 · doi:10.1080/00207160.2010.521550
[36]Zuzana, K.; Karol, M.: An adaptive finite volume scheme for solving nonlinear diffusion equations in image processing, J. visual commun. Image representation 13, No. 1, 22-35 (2002)