Two-Grid Method for Nonlinear Reaction-Diffusion Equations by Mixed Finite Element Methods

In this paper, we investigate a scheme for nonlinear reaction-diffusion equations using the mixed finite element methods. To linearize the mixed method equations, we use the two-grid algorithm. First, we solve the original nonlinear equations on the coarse grid, then, we solve the linearized problem on the fine grid used Newton iteration once. It is shown that the algorithm can achieve asymptotically optimal approximation as long as the mesh sizes satisfy $H=\mathcal{O}(h^{\frac{1}{2}})$. As a result, solving such a large class of nonlinear equations will not much more difficult than the solution of one linearized equation.

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

[2]  Yanping Chen,et al.  Analysis of Two-Grid Methods for Nonlinear Parabolic Equations by Expanded Mixed Finite Element Methods , 2009 .

[3]  P. Raviart,et al.  A mixed finite element method for 2-nd order elliptic problems , 1977 .

[4]  G. Pinder,et al.  Computational Methods in Subsurface Flow , 1983 .

[5]  M. Allen,et al.  A two-grid method for mixed finite-element solution of reaction-diffusion equations , 1999 .

[6]  M. Wheeler,et al.  Mixed Finite Elements for Elliptic Problems with Tensor Coefficients as Cell-Centered Finite Differences , 1997 .

[7]  Li Li,et al.  Lp error estimates of two-grid schemes of expanded mixed finite element methods , 2009, Appl. Math. Comput..

[8]  Yanping Chen,et al.  Analysis of two‐grid methods for reaction‐diffusion equations by expanded mixed finite element methods , 2007 .

[9]  Jean E. Roberts,et al.  Global estimates for mixed methods for second order elliptic equations , 1985 .

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

[11]  M. Cristina,et al.  Superconvergence of mixed finite element methods for parabolic equations , 1987 .

[12]  L. D. Marini,et al.  Two families of mixed finite elements for second order elliptic problems , 1985 .

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

[14]  陈娜,et al.  一类考虑接种的结核病模型的稳定性分析 Dynamics of a TB Model with Vaccination , 2012 .

[15]  Miente Bakker.  Aspects of the finite element method , 1982 .

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

[17]  Jianfeng Zhu,et al.  Application of Discontinuous Galerkin Methods for Reaction-Diffusion Systems in Developmental Biology , 2009, J. Sci. Comput..

[18]  Sonia M. F. Garcia.  Improved error estimates for mixed finite‐element approximations for nonlinear parabolic equations: The continuous‐time case , 1994 .

[19]  Eun-Jae Park,et al.  Mixed finite element methods for nonlinear second-order elliptic problems , 1995 .

[20]  Richard E. Ewing,et al.  The approximation of the pressure by a mixed method in the simulation of miscible displacement , 1983 .

[21]  V. Thomée.  Galerkin Finite Element Methods for Parabolic Problems (Springer Series in Computational Mathematics) , 2010 .

[22]  Yunqing Huang,et al.  A two‐grid method for expanded mixed finite‐element solution of semilinear reaction–diffusion equations , 2003 .