L2-Stability Independent of Diffusion for a Finite Element-Finite Volume Discretization of a Linear Convection-Diffusion Equation

We consider a time-dependent and a steady linear convection-diffusion equation. These equations are approximately solved by a combined finite element--finite volume method: the diffusion term is discretized by Crouzeix--Raviart piecewise linear finite elements on a triangular grid, and the convection term by upwind barycentric finite volumes. In the unsteady case, the implicit Euler method is used as time discretization. This scheme is shown to be unconditionally $L^2$-stable, uniformly with respect to the diffusion coefficient.

[1]  Miloslav Feistauer,et al.  Error Estimates for Barycentric Finite Volumes Combined with Nonconforming Finite Elements Applied to Nonlinear Convection-Diffusion Problems , 2002 .

[2]  Erik Burman,et al.  Finite element methods with symmetric stabilization for the transient convection―diffusion-reaction equation , 2009 .

[3]  Zhitao Li.  Convergence analysis of an upwind mixed element method for advection diffusion problems , 2009, Appl. Math. Comput..

[4]  Marc Buffat,et al.  An implicit mixed finite‐volume–finite‐element method for solving 3D turbulent compressible flows , 1997 .

[5]  Aijie Cheng,et al.  A uniformly optimal‐order estimate for finite volume method for advection‐diffusion equations , 2014 .

[6]  Miloslav Feistauer,et al.  Analysis of space–time discontinuous Galerkin method for nonlinear convection–diffusion problems , 2011, Numerische Mathematik.

[7]  Regina Leal-Toledo,et al.  Numerical analysis of a least-squares finite element method for the time-dependent advection-diffusion equation , 2011, J. Comput. Appl. Math..

[8]  Martin Vohralík,et al.  A combined finite volume–finite element scheme for the discretization of strongly nonlinear convection–diffusion–reaction problems on nonmatching grids , 2009 .

[9]  Thierry Gallouët,et al.  A convergent finite element-finite volume scheme for the compressible Stokes problem. Part I: The isothermal case , 2007, Math. Comput..

[10]  Riccardo Sacco,et al.  Stabilized Mixed Finite Volume Methods for Convection-Diffusion Problems , 1997 .

[11]  J. Guermond,et al.  Theory and practice of finite elements , 2004 .

[12]  Martin Vohralík,et al.  A combined finite volume–nonconforming/mixed-hybrid finite element scheme for degenerate parabolic problems , 2006, Numerische Mathematik.

[13]  Philippe Angot,et al.  Analysis of a combined barycentric finite volume—nonconforming finite element method for nonlinear convection-diffusion problems , 1998 .

[14]  Gabriel R. Barrenechea,et al.  A local projection stabilization finite element method with nonlinear crosswind diffusion for convection-diffusion-reaction equations , 2012 .

[15]  Todd Arbogast,et al.  A Nonlinear Mixed Finite Eelement Method for a Degenerate Parabolic Equation Arising in Flow in Porous Media , 1996 .

[16]  Marcus Mildner,et al.  Stabilité de l'équation d'advection-diffusion et stabilité de l'équation d'advection pour la solution du problème approché, obtenue par la méthode upwind d'éléments-finis et de volumes-finis avec des éléments de Crouzeix-Raviart , 2013 .

[17]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[18]  Paul Deuring,et al.  Stability of a combined finite element ‐ finite volume discretization of convection‐diffusion equations , 2012 .

[19]  Miloslav Feistauer,et al.  On the convergence of a combined finite volume–finite element method for nonlinear convection–diffusion problems. Explicit schemes , 1999 .

[20]  Rüdiger Verfürth,et al.  Robust A Posteriori Error Estimates for Stationary Convection-Diffusion Equations , 2005, SIAM J. Numer. Anal..

[21]  Thierry Gallouët,et al.  Convergence of an upstream finite volume scheme for a nonlinear hyperbolic equation on a triangular mesh , 1993 .

[22]  Petr Knobloch.  UNIFORM VALIDITY OF DISCRETE FRIEDRICHS' INEQUALITY FOR GENERAL NONCONFORMING FINITE ELEMENT SPACES , 2001 .

[23]  A. Serghini Mounim,et al.  A stabilized finite element method for convection–diffusion problems , 2012 .

[24]  Blanca Ayuso de Dios,et al.  Discontinuous Galerkin Methods for Advection-Diffusion-Reaction Problems , 2009, SIAM J. Numer. Anal..

[25]  M. Lukáčová-Medvid'ová,et al.  On the Convergence of a Combined Finite Volume{Finite Element Method for Nonlinear Convection{Diffusion Problems , 1997 .

[26]  A. Friedman.  Partial Differential Equations of Parabolic Type , 1983 .

[27]  Katsushi Ohmori,et al.  A technique of upstream type applied to a linear nonconforming finite element approximation of convective diffusion equations , 1984 .

[28]  Giancarlo Sangalli,et al.  Analysis of a Multiscale Discontinuous Galerkin Method for Convection-Diffusion Problems , 2006, SIAM J. Numer. Anal..

[29]  Thierry Gallouët,et al.  A convergent finite element-finite volume scheme for the compressible Stokes problem. Part II: the isentropic case , 2009, Math. Comput..

[30]  P. Causin,et al.  FLUX-UPWIND STABILIZATION OF THE DISCONTINUOUS PETROV-GALERKIN FORMULATION WITH LAGRANGE MULTIPLIERS FOR ADVECTION-DIFFUSION PROBLEMS ∗ , 2005 .

[31]  Alain Dervieux,et al.  Computation of unsteady flows with mixed finite volume/finite element upwind methods , 1998 .

[32]  P. Bassanini,et al.  Elliptic Partial Differential Equations of Second Order , 1997 .

[33]  Susanne C. Brenner,et al.  Poincaré-Friedrichs Inequalities for Piecewise H1 Functions , 2003, SIAM J. Numer. Anal..

[34]  M. Vohralík.  On the Discrete Poincaré–Friedrichs Inequalities for Nonconforming Approximations of the Sobolev Space H 1 , 2005 .

[35]  Paul Deuring,et al.  Error estimates for a finite element-finite volume discretization of convection-diffusion equations , 2011 .

[36]  Po-Wen Hsieh,et al.  On efficient least-squares finite element methods for convection-dominated problems , 2009 .

[37]  Sabine Fenstermacher,et al.  Numerical Approximation Of Partial Differential Equations , 2016 .

[38]  Douglas Enright,et al.  Mathematical and computational methods for compressible flow , 2005, Math. Comput..

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

[40]  L. Tobiska,et al.  A nonconforming finite element method of upstream type applied to the stationary Navier-Stokes equation , 1989 .

[41]  Miloslav Feistauer,et al.  Mathematical Methods in Fluid Dynamics , 1993 .

[42]  Vít Dolejší,et al.  On the Discrete Friedrichs Inequality for Nonconforming Finite Elements , 1999 .

[43]  Anders Szepessy,et al.  Convergence of a streamline diffusion finite element method for a conservation law with boundary conditions , 1991 .