The semidiscrete finite volume element method for nonlinear convection-diffusion problem

Abstract In this work, a semidiscrete finite volume element method for the nonlinear convection–diffusion problem is considered. Under some regular assumptions, we obtain the L2 and H1 norm error estimates of numerical solution. Furthermore, we investigate two-grid finite volume element method for the considered equations. Compared with the standard method, the two-grid method is of the same order as the standard method in the H1-norm as long as the mesh sizes satisfy H = O h 1 3 . However, the two-grid method involves much less work than the standard method. Finally, some numerical results are provided to verify the established theoretical analysis.

[1]  C. Schwab,et al.  A finite volume discontinuous Galerkin scheme¶for nonlinear convection–diffusion problems , 2002 .

[2]  Chuanjun Chen,et al.  Two-grid methods for finite volume element approximations of nonlinear parabolic equations , 2009 .

[3]  Jinchao Xu,et al.  A Novel Two-Grid Method for Semilinear Elliptic Equations , 1994, SIAM J. Sci. Comput..

[4]  Karol Mikula,et al.  Semi-implicit finite volume scheme for solving nonlinear diffusion equations in image processing , 2001, Numerische Mathematik.

[5]  William Layton,et al.  Two-level Picard and modified Picard methods for the Navier-Stokes equations , 1995 .

[6]  Mary F. Wheeler,et al.  A Two-Grid Finite Difference Scheme for Nonlinear Parabolic Equations , 1998 .

[7]  L. R. Scott,et al.  The Mathematical Theory of Finite Element Methods , 1994 .

[8]  Mary F. Wheeler,et al.  Two-grid methods for mixed finite element approxi-mations of nonlinear parabolic equations , 1994 .

[9]  Li Ronghua,et al.  Generalized difference methods for a nonlinear Dirichlet problem , 1987 .

[10]  Victor Ginting,et al.  Two-grid finite volume element method for linear and nonlinear elliptic problems , 2007, Numerische Mathematik.

[11]  Jing Zhao,et al.  A full discrete two-grid finite-volume method for a nonlinear parabolic problem , 2011, Int. J. Comput. Math..

[12]  J. Varah.  Stability Restrictions on Second Order, Three Level Finite Difference Schemes for Parabolic Equations , 1978 .

[13]  Ronghua Li.  Generalized Difference Methods for Differential Equations: Numerical Analysis of Finite Volume Methods , 2000 .

[14]  Haijun Wu,et al.  Error estimates for finite volume element methods for general second‐order elliptic problems , 2003 .

[15]  Endre Süli,et al.  Approximation of the global attractor for the incompressible Navier–Stokes equations , 2000 .

[16]  高夫征.  Finite Volume Element Predictor-corrector Method for a Class of Nonlinear Parabolic Systems , 2005 .

[17]  Jim E. Jones,et al.  Control‐volume mixed finite element methods , 1996 .

[18]  Jinchao Xu,et al.  Analysis of linear and quadratic simplicial finite volume methods for elliptic equations , 2009, Numerische Mathematik.

[19]  Qian Li,et al.  Error estimates in L2, H1 and Linfinity in covolume methods for elliptic and parabolic problems: A unified approach , 1999, Math. Comput..

[20]  Olga Stasová,et al.  Convergence Analysis of Finite Volume Scheme for Nonlinear Tensor Anisotropic Diffusion in Image Processing , 2007, SIAM J. Numer. Anal..

[21]  Tao Lin,et al.  On the Accuracy of the Finite Volume Element Method Based on Piecewise Linear Polynomials , 2001, SIAM J. Numer. Anal..

[22]  Jinchao Xu.  Two-grid Discretization Techniques for Linear and Nonlinear PDEs , 1996 .

[23]  Eugene O'Riordan,et al.  Special meshes for finite difference approximations to an advection-diffusion equation with parabolic layers , 1995 .

[24]  Yinnian He,et al.  Finite volume method based on stabilized finite elements for the nonstationary Navier–Stokes problem , 2007 .

[25]  Zhiqiang Cai,et al.  The finite volume element method for diffusion equations on general triangulations , 1991 .

[26]  Richard E. Ewing,et al.  Some new error estimates of a semidiscrete finite volume element method for a parabolic integro-differential equation with nonsmooth initial data , 2006, SIAM J. Numer. Anal..

[27]  R. Eymard,et al.  Finite Volume Methods , 2019, Computational Methods for Fluid Dynamics.

[28]  Jinchao Xu,et al.  Error estimates on a new nonlinear Galerkin method based on two-grid finite elements , 1995 .

[29]  M. Stynes,et al.  Numerical methods for singularly perturbed differential equations : convection-diffusion and flow problems , 1996 .

[30]  Richard E. Ewing,et al.  A Modified Finite Volume Approximation of Second-Order Elliptic Equations with Discontinuous Coefficients , 2001, SIAM J. Sci. Comput..

[31]  Yirang Yuan,et al.  The characteristic finite volume element method for the nonlinear convection-dominated diffusion problem , 2008, Comput. Math. Appl..

[32]  Gerald Warnecke,et al.  Error Estimates for a Combined Finite Volume--Finite Element Method for Nonlinear Convection--Diffusion Problems , 1999 .

[33]  D. Rose,et al.  Some errors estimates for the box method , 1987 .

[34]  Gao Pu-zheng.  Finite Volume Element Predictor-corrector Method for a Class of Nonlinear Parabolic Systems , 2005 .

[35]  Karol Mikula,et al.  An Adaptive Finite Volume Scheme for Solving Nonlinear Diffusion Equations in Image Processing , 2002, J. Vis. Commun. Image Represent..