The lowest order characteristic mixed finite element scheme for convection-dominated diffusion problem

A new characteristic mixed finite element method (MFEM) is proposed by combining the method of characteristics with the new mixed variational formulation for the convection-dominated diffusion problem. The error estimates for both the original variable u and auxiliary variable p ? are derived via the properties of the integral identity and mean value techniques. At the same time, some numerical experiments are provided to confirm validity of the theoretical analysis and excellent performance of the proposed method.

[1]  Chen Fengxin.  AN EXPANDED CHARACTERISTICS-MIXED FINITE ELEMENT METHOD FOR QUASILINEAR CONVECTION-DOMINATED DIFFUSION EQUATIONS , 2009 .

[2]  Fawang Liu,et al.  Numerical Methods for the Variable-Order Fractional Advection-Diffusion Equation with a Nonlinear Source Term , 2009, SIAM J. Numer. Anal..

[3]  T. F. Russell,et al.  NUMERICAL METHODS FOR CONVECTION-DOMINATED DIFFUSION PROBLEMS BASED ON COMBINING THE METHOD OF CHARACTERISTICS WITH FINITE ELEMENT OR FINITE DIFFERENCE PROCEDURES* , 1982 .

[4]  Dan-ping Yang,et al.  Some least-squares Galerkin procedures for first-order time-dependent convection–diffusion system , 1999 .

[5]  Xinlong Feng,et al.  A new mixed finite element method based on the Crank–Nicolson scheme for the parabolic problems , 2012 .

[6]  HONG WANG,et al.  Uniform Estimates for Eulerian-Lagrangian Methods for Singularly Perturbed Time-Dependent Problems , 2007, SIAM J. Numer. Anal..

[7]  Dongyang Shi,et al.  High accuracy analysis of the characteristic‐nonconforming FEM for a convection‐dominated transport problem , 2014 .

[8]  Yadong Zhang,et al.  A New Characteristic Nonconforming Mixed Finite Element Scheme for Convection-Dominated Diffusion Problem , 2013, J. Appl. Math..

[9]  Xinlong Feng,et al.  The semi-discrete streamline diffusion finite element method for time-dependented convection-diffusion problems , 2008, Appl. Math. Comput..

[10]  M. Wheeler,et al.  A characteristics-mixed finite element method for advection-dominated transport problems , 1995 .

[11]  Hui Guo,et al.  A split least-squares characteristic mixed element method for nonlinear nonstationary convection–diffusion problem , 2012, Int. J. Comput. Math..

[12]  Wei Liu,et al.  A two‐grid characteristic finite volume element method for semilinear advection‐dominated diffusion equations , 2013 .

[13]  Chieh-Sen Huang,et al.  The modified method of characteristics with adjusted advection , 1999, Numerische Mathematik.

[14]  Gianmarco Manzini,et al.  A finite volume method for advection-diffusion problems in convection-dominated regimes , 2008 .

[15]  Philippe G. Ciarlet,et al.  The finite element method for elliptic problems , 2002, Classics in applied mathematics.

[16]  Liao Kai-tai.  A New Mixed Finite Element Scheme for Elliptic Equations , 2011 .

[17]  Fengxin Chen,et al.  Characteristic mixed finite element approximation of transient convection diffusion optimal control problems , 2012, Math. Comput. Simul..

[18]  Peter Hansbo,et al.  The characteristic streamline diffusion method for convection-diffusion problems , 1992 .

[19]  Yadong Zhang,et al.  High accuracy analysis of a new nonconforming mixed finite element scheme for Sobolev equations , 2011, Appl. Math. Comput..

[20]  Lin Qun.  A Uniform Estimate for the Characteristics Bilinear Element Method for Advection-Diffusion Equations , 2011 .

[21]  Dongyang Shi,et al.  Low Order Nonconforming Expanded Characteristic- Mixed Finite Element Method for the Convection- Diffusion Problem , 2013 .

[22]  Zhangxin Chen,et al.  Reliable and efficient error control for an adaptive Galerkin-characteristic method for convection-dominated diffusion problems , 2012, Adv. Comput. Math..

[23]  Chen,et al.  AN EXPANDED CHARACTERISTIC-MIXED FINITE ELEMENT METHOD FOR A CONVECTION-DOMINATED TRANSPORT PROBLEM , 2005 .

[24]  Fawang Liu,et al.  A characteristic difference method for the variable-order fractional advection-diffusion equation , 2013 .

[25]  J. Nédélec.  A new family of mixed finite elements in ℝ3 , 1986 .

[26]  Dong-Yang Shi,et al.  Two low order characteristic finite element methods for a convection-dominated transport problem , 2010, Comput. Math. Appl..

[27]  Haitao Che,et al.  A characteristics-mixed covolume method for a convection-dominated transport problem , 2009, J. Comput. Appl. Math..

[28]  T. F. Russell,et al.  An Eulerian-Lagrangian localized adjoint method for the advection-diffusion equation , 1990 .

[29]  Zhangxin Chen,et al.  Characteristic-nonconforming finite-element methods for advection-dominated diffusion problems , 2004 .

[30]  Jure Ravnik,et al.  A gradient free integral equation for diffusion–convection equation with variable coefficient and velocity , 2013 .

[31]  G. M.,et al.  Partial Differential Equations I , 2023, Applied Mathematical Sciences.

[32]  Jure Ravnik,et al.  Integral equation formulation of an unsteady diffusion-convection equation with variable coefficient and velocity , 2014, Comput. Math. Appl..