We consider a time-dependent and a steady linear convection-diffusion-reaction equation whose coefficients are nonconstant. Boundary conditions are mixed (Dirichlet and Robin−Neumann) and nonhomogeneous. Both the unsteady and the steady problem 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 -stable, uniformly with respect to diffusion, except if the Robin−Neumann boundary condition is inhomogeneous and the convective velocity is tangential at some points of the Robin−Neumann boundary. In that case, a negative power of the diffusion coefficient arises. As is shown by a counterexample, this exception cannot be avoided.
Accepté le :
DOI : 10.1051/m2an/2016042
Mots-clés : Convection-diffusion equation, combined finite element – finite volume method, Crouzeix–Raviart finite elements, barycentric finite volumes, upwind method, stability
@article{M2AN_2017__51_3_919_0, author = {Deuring, Paul and Eymard, Robert}, title = {$L^{2}$-stability of a finite element {\textendash} finite volume discretization of convection-diffusion-reaction equations with nonhomogeneous mixed boundary conditions}, journal = {ESAIM: Mathematical Modelling and Numerical Analysis }, pages = {919--947}, publisher = {EDP-Sciences}, volume = {51}, number = {3}, year = {2017}, doi = {10.1051/m2an/2016042}, mrnumber = {3666651}, zbl = {1371.65096}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/m2an/2016042/} }
TY - JOUR AU - Deuring, Paul AU - Eymard, Robert TI - $L^{2}$-stability of a finite element – finite volume discretization of convection-diffusion-reaction equations with nonhomogeneous mixed boundary conditions JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2017 SP - 919 EP - 947 VL - 51 IS - 3 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/m2an/2016042/ DO - 10.1051/m2an/2016042 LA - en ID - M2AN_2017__51_3_919_0 ER -
%0 Journal Article %A Deuring, Paul %A Eymard, Robert %T $L^{2}$-stability of a finite element – finite volume discretization of convection-diffusion-reaction equations with nonhomogeneous mixed boundary conditions %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2017 %P 919-947 %V 51 %N 3 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/m2an/2016042/ %R 10.1051/m2an/2016042 %G en %F M2AN_2017__51_3_919_0
Deuring, Paul; Eymard, Robert. $L^{2}$-stability of a finite element – finite volume discretization of convection-diffusion-reaction equations with nonhomogeneous mixed boundary conditions. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 3, pp. 919-947. doi : 10.1051/m2an/2016042. http://archive.numdam.org/articles/10.1051/m2an/2016042/
R.A. Adams, Sobolev Spaces. Academic Press, New York e.a. (1975). | MR | Zbl
Analysis of a combined barycentric finite volume – nonconforming finite element method for nonlinear convection-diffusion problems. Appl. Math. 43 (1998) 263–310. | DOI | MR | Zbl
, , and ,Discontinuous Galerkin methods for advection-diffusion-reaction problems. SIAM J. Numer. Anal. 47 (2009) 1391–1420. | DOI | MR | Zbl
and ,A local projection stabilization finite element method with nonlinear crosswind diffusion for convection-diffusion-reaction equation. Math. Model. Numer. Anal. 47 (2013) 1335–1366. | DOI | Numdam | MR | Zbl
, and ,Combined triangular FV-triangular FE method for nonlinear convection-diffusion problems. Z. Angew. Math. Mech. 87 (2007) 499–517. | DOI | MR | Zbl
, , , and ,S.C. Brenner, The Mathematical Theory of Finite Element Methods (2nd ed.). Springer, New York e.a. (2002). | Zbl
Poincaré-Friedrichs inequalities for piecewise -functions. SIAM J. Numer. Anal. 41 (2003) 306–324. | DOI | MR | Zbl
,Analysis of multiscale discontinuous Galerkin method for convection-diffusion problems. SIAM J. Numer. Anal. 44 (2006) 1420–1440. | DOI | MR | Zbl
, and ,Finite element methods with symmetric stabilization for the transient convection-diffusion-reaction equation. Comput. Methods Appl. Mech. Engrg. 198 (2009) 2508–2519. | DOI | MR | Zbl
and ,Flux-upwind stabilization of the discontinuous Petrov-Galerkin formulation with Lagrangian multipliers for advection-diffusion problems. ESAIM: M2AN 39 (2005) 1087–1114. | DOI | Numdam | MR | Zbl
, and ,P.G. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1979). | MR | Zbl
P. Deuring, A finite element – finite volume discretization of convection-reaction-diffusion equations with mixed nonhomogeneous boundary conditions: error estimates. To appear in Numer. Methods Partial Differ. Equ. | MR
Error estimates for a finite element – finite volume discretization of convection-diffusion equations. Appl. Numer. Math. 61 (2011) 785–801. | DOI | MR | Zbl
and ,Stability of a combined finite element – finite volume discretization of convection-diffusion equations. Numer. Methods Partial Differ. Equ. 28 (2012) 402–424. | DOI | MR | Zbl
and ,L-stability independent of diffusion for a finite element – finite volume discretization of a linear convection-diffusion equation. SIAM J. Numer. Anal. 53 (2015) 508–526. | DOI | MR | Zbl
, and ,The asymptotic behavior of the first real eigenvalue of second order elliptic operators with a small parameter in the highest derivative. II. Indiana Univ. Math. J. 23 (1973) 991–1011. | DOI | MR | Zbl
, and ,On the discrete Friedrichs inequality for non-conforming finite elements. Numer. Funct. Anal. Optim. 20 (1999) 437–447. | DOI | MR | Zbl
, and ,Error estimates for barycentric finite volumes combined with nonconforming finite elements applied to nonlinear convection-diffusion problems. Appl. Math. 47 (2002) 301–340. | DOI | MR | Zbl
, , and ,A. Ern and J.-L. Guermond, Theory and Practice of Finite Elements. Appl. Math. Sci. 159, Springer, New York e.a. (2004). | MR | Zbl
A convergent finite element - finite volume scheme for the compressible Stokes problem. Part II: the isentropic case. Math. Comput. 270 649–675 (2010). | MR | Zbl
, , and ,A combined finite volume-nonconforming/mixed-hybrid finite element scheme for degenerate parabolic problems. Numer. Math. 105 (2006) 73–131. | DOI | MR | Zbl
, and ,A combined finite volume – finite element scheme for the discretization of strongly nonlinear convection-diffusion-reaction problems on nonmatching grids. Numer. Methods Partial Differ. Equ. 26 (2010) 612–646. | MR | Zbl
, and ,M. Feistauer, Mathematical Methods in Fluid Dynamics. Pitman Monographs and Surveys in Pure and Applied Mathematics 67. Longman, Harlow (1993). | MR | Zbl
On the convergence of a combined finite volume – finite element method for nonlinear convection-diffusion problems. Numer. Methods Partial Differ. Equ. 13 (1997) 163–190. | DOI | MR | Zbl
, and ,Error estimates for a combined finite volume – finite element method for nonlinear convection-diffusion problems. SIAM J. Math. Anal. 36 (1999) 1528–1548. | MR | Zbl
, , and ,M. Feistauer, J. Felcman and I. Straškraba, Mathematical and Computational Methods for Compressible Flow. Clarendon Press, Oxford (2003). | MR | Zbl
Analysis of a space-time discontinuous Galerkin method for nonlinear convection-diffusion problems. Numer. Math. 117 (2011) 251–288. | DOI | MR | Zbl
, , and ,On the convergence of a combined finite volume – finite element method for nonlinear convection-diffusion problems. Explicit schemes. Numer. Methods Partial Differ. Equ. 15 (1999) 215–235. | DOI | MR | Zbl
, and ,S. Fučík, O. John and A. Kufner, Function Spaces. Noordhoff, Leyden (1977). | MR | Zbl
A convergent finite element – finite volume scheme for the compressible Stokes problem. Part I: the isothermal case. Math. Comput. 267 (2009) 1333–1352. | DOI | MR | Zbl
, and ,V. Girault and P.-A. Raviart, Finite Element Methods for Navier–Stokes Equations. Springer, Berlin e.a. (1986). | MR | Zbl
P. Grisvard, Elliptic Problems in Nonsmooth Domains. Pitman, London (1985). | MR | Zbl
An implicit mixed finite-volume-finite-element method for solving 3D turbulent compressible flows. Int. J. Numer. Meth. Fluids 25 (1997) 1241–1261. | DOI | Zbl
, and ,On efficient least-squares finite element methods for convection-dominated problems. Comput. Methods Appl. Mech. Engrg. 199 (2009) 183–196. | DOI | MR | Zbl
and ,A. Jonsson and H. Wallin, Function Spaces on Subsets of . Harwood Academic Publishers, New York (1984). | MR | Zbl
N. Kikuchi and J.T. Oden, Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods. SIAM, Philadelphia (1988). | MR | Zbl
Uniform validity of discrete Friedrich’s inequality for general nonconforming finite element spaces. Numer. Funct. Anal. Optimiz. 22 (2001) 107–126. | DOI | MR | Zbl
,Numerical analysis of a least-squares finite element method for the time-dependent advection-diffusion equation. J. Comp. Appl. Math. 235 (2011) 3615–3631. | DOI | MR | Zbl
and ,Convergence analysis of an upwind mixed element method for advection diffusion problems. Appl. Math. Comput. 212 (2009) 318–326. | MR | Zbl
,A stabilized finite element method for convection-diffusion problems. Numer. Methods Partial Differ. Equ. 28 (2012) 1916–1943. | DOI | MR | Zbl
,A technique of upstream type applied to a linear nonconforming finite element approximation of convective diffusion equations. RAIRO Modél. Math. Anal. Numér. 18 (1984) 309–332. | DOI | Numdam | MR | Zbl
and ,A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations. Springer Series in Computational Mathematics 23. Springer, Berlin e.a. (1994). | MR | Zbl
A uniformly optimal-order estimate for finite volume method for advection-diffusion equation. Numer. Methods Partial Differ. Equ. 30 (2014) 17–43. | DOI | MR | Zbl
, and ,H.-G. Roos, M. Stynes and L. Tobiska, Robust Numerical Methods for Singularly Perturbed Differential Equations (2nd ed.). Springer Series in Computational Mathematics 24. Springer, New York e.a. (2008). | MR | Zbl
H.L. Royden, Real Analysis. Macmillan, New York (1968). | MR | Zbl
R. Temam, Navier-Stokes Equations. North-Holland, Amsterdam (1977). | MR | Zbl
H. Triebel, Interpolation Theory, Function Spaces, Differential Operators. North Holland, Amsterdam e.a. (1978). | MR | Zbl
Robust a posteriori error estimates for steady convection-diffusion equations. SIAM J. Numer. Anal. 43 (2005) 1766–1782. | DOI | MR | Zbl
,Robust a posteriori error estimates for unsteady convection-diffusion equations. SIAM J. Numer. Anal. 43 (2005) 1783–1802. | DOI | MR | Zbl
,A priori error estimates of an extrapolated space-time discontinuous Galerkin method for nonlinear convection-diffusion problems. Numer. Methods Partial Differ. Equ. 27 (2011) 1456–1482. | DOI | MR | Zbl
, and ,On the discrete Poincaré-Friedrichs inequalities for nonconforming approximations of the Sobolev space . Numer. Funct. Anal. Optimiz. 26 (2005) 925–952. | DOI | MR | Zbl
,J. Wloka, Partial Differential Equations. Cambridge University Press, Cambridge e.a. (1987). | MR | Zbl
Cité par Sources :