Error estimates for a finite element-finite volume discretization of convection-diffusion equations

We consider a time-dependent linear convection-diffusion equation. This equation is approximated by a combined finite element-finite volume method: the diffusion term is discretized by Crouzeix-Raviart piecewise linear finite elements, and the convection term by upwind barycentric finite volumes on a triangular grid. An implicit Euler approach is used for time discretization. It is shown that the error associated with this scheme, measured by a discrete L^~-L^2- and L^2-H^1-norm, respectively, decays linearly with the mesh size and the time step. This result holds without any link between mesh size and time step. The dependence of the corresponding error bound on the diffusion coefficient is completely explicit.

[1]  M. Tidriri.  Error estimates for the hybrid finite element/finite volume methods for linear hyperbolic and convection-dominated problems , 2003 .

[2]  Tsutomu Ikeda,et al.  Maximum Principle in Finite Element Models for Convection-diffusion Phenomena , 1983 .

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

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

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

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

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

[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]  R. Temam.  Navier-Stokes Equations , 1977 .

[10]  J. Wloka,et al.  Partial differential equations , 1987 .

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

[12]  Stefano Micheletti,et al.  Stability and error analysis of mixed finite-volume methods for advection dominated problems , 2006, Comput. Math. Appl..

[13]  M. Tidriri.  Analysis of the hybrid finite element/finite volume methods for linear hyperbolic and convection-dominated convection-diffusion problems , 2002 .

[14]  P. Grisvard.  Elliptic Problems in Nonsmooth Domains , 1985 .

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

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

[17]  H. Rui.  Convergence of an upwind control-volume mixed finite element method for convection–diffusion problems , 2007, Computing.

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

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

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

[21]  Miloslav Feistauer,et al.  Mathematical and Computational Methods for Compressible Flow , 2003 .

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

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

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

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

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

[27]  Song Wang,et al.  A nonconforming combination of the finite element and volume methods with an anisotropic mesh refinement for a singularly perturbed convection-diffusion equation , 2003, Math. Comput..

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

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

[30]  Miloslav Feistauer,et al.  L ∞ (L 2)-error estimates for the DGFEM applied to convection–diffusion problems on nonconforming meshes , 2009, J. Num. Math..

[31]  A. Quarteroni,et al.  Numerical Approximation of Partial Differential Equations , 2008 .