Numerical analysis of a least-squares finite element method for the time-dependent advection-diffusion equation

A mixed finite element scheme designed for solving the time-dependent advection-diffusion equations expressed in terms of both the primal unknown and its flux, incorporating or not a reaction term, is studied. Once a time discretization of the Crank-Nicholson type is performed, the resulting system of equations allows for a stable approximation of both fields, by means of classical Lagrange continuous piecewise polynomial functions of arbitrary degree, in any space dimension. Convergence in the norm of H^1xH(div) in space and in appropriate senses in time applying to this pair of fields is demonstrated.

[1]  Mutsuto Kawahara,et al.  A finite element method for high Reynolds number viscous fluid flow using two step explicit scheme , 1983 .

[2]  Hiroshi Fujita.  Operator Theory and Numerical Methods , 2011 .

[3]  J. H. Carneiro de Araujo,et al.  Study of a finite element method for the time-dependent generalized Stokes system associated with viscoelastic flow , 2010, J. Comput. Appl. Math..

[4]  Vivette Girault,et al.  Finite Element Methods for Navier-Stokes Equations - Theory and Algorithms , 1986, Springer Series in Computational Mathematics.

[5]  Masahisa Tabata,et al.  On a conservation upwind finite element scheme for convective diffusion equations , 1981 .

[6]  O. C. Zienkiewicz,et al.  An ‘upwind’ finite element scheme for two‐dimensional convective transport equation , 1977 .

[7]  G. Strang,et al.  An Analysis of the Finite Element Method , 1974 .

[8]  T. Hughes,et al.  Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations , 1990 .

[9]  A. Brasil,et al.  An explicit method for convection-diffusion equations , 2009 .

[10]  R. D. Richtmyer,et al.  Survey of the stability of linear finite difference equations , 1956 .

[11]  Joseph E. Pasciak,et al.  A least-squares approach based on a discrete minus one inner product for first order systems , 1997, Math. Comput..

[12]  T. Manteuffel,et al.  FIRST-ORDER SYSTEM LEAST SQUARES FOR SECOND-ORDER PARTIAL DIFFERENTIAL EQUATIONS : PART II , 1994 .

[13]  David K. Gartling.  Some comments on the paper by Heinrich, Huyakorn, Zienkiewicz and Mitchell , 1978 .

[14]  J. H. Carneiro de Araujo,et al.  A quadratic triangle of the Hermite type for second order elliptic problems , 2009 .

[15]  G. Carey,et al.  Least-squares mixed finite elements for second-order elliptic problems , 1994 .

[16]  P. Grisvard.  Singularities in Boundary Value Problems , 1992 .

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