$L^{2}$-stability of a finite element – finite volume discretization of convection-diffusion-reaction equations with nonhomogeneous mixed boundary conditions

Deuring, Paul; Eymard, Robert

doi:10.1051/m2an/2016042

L^{2}

-stability of a finite element – finite volume discretization of convection-diffusion-reaction equations with nonhomogeneous mixed boundary conditions

Deuring, Paul ¹ ; Eymard, Robert ²

¹ Universitédu Littoral Côte d’Opale, Laboratoire de mathématiques pures et appliquées Joseph Liouville, 62228 Calais, France.
² UniversitéParis-Est Marne-la-Vallée, 5 boulevard Descartes, 77454 Marne-la-Vallée, France.

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 3, pp. 919-947.

Résumé

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 $L^{2}$ -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.

Reçu le : 2015-09-09
Accepté le : 2016-05-30

MR Zbl

DOI : 10.1051/m2an/2016042

Classification : 65M12, 65M60
Mots clés : Convection-diffusion equation, combined finite element – finite volume method, Crouzeix–Raviart finite elements, barycentric finite volumes, upwind method, stability

Affiliations des auteurs :

Deuring, Paul ¹ ; Eymard, Robert ²

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     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},
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     mrnumber = {3666651},
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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/

Bibliographie
Cité par

R.A. Adams, Sobolev Spaces. Academic Press, New York e.a. (1975). | MR | Zbl

P. Angot, V. Dolejší, M. Feistauer and J. Felcman, 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

B. Ayuso and L.D. Marini, Discontinuous Galerkin methods for advection-diffusion-reaction problems. SIAM J. Numer. Anal. 47 (2009) 1391–1420. | DOI | MR | Zbl

G.R. Barrenechea, V. John and P. Knobloch, 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

M. Bejček, M. Feistauer, T. Gallouët, J. Hájek and R. Herbin, Combined triangular FV-triangular FE method for nonlinear convection-diffusion problems. Z. Angew. Math. Mech. 87 (2007) 499–517. | DOI | MR | Zbl

S.C. Brenner, The Mathematical Theory of Finite Element Methods (2nd ed.). Springer, New York e.a. (2002). | Zbl

S.C. Brenner, Poincaré-Friedrichs inequalities for piecewise $H^{1}$ -functions. SIAM J. Numer. Anal. 41 (2003) 306–324. | DOI | MR | Zbl

A. Buffa, T.J.R. Hughes and G. Sangalli, Analysis of multiscale discontinuous Galerkin method for convection-diffusion problems. SIAM J. Numer. Anal. 44 (2006) 1420–1440. | DOI | MR | Zbl

E. Burman and M. A. Fernández, 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

P. Causin, R. Sacco and C.L. Bottasso, 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

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

P. Deuring and M. Mildner, Error estimates for a finite element – finite volume discretization of convection-diffusion equations. Appl. Numer. Math. 61 (2011) 785–801. | DOI | MR | Zbl

P. Deuring and M. Mildner, 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

P. Deuring, R. Eymard and M. Mildner, L $^{2}$ -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

A. Devinatz, R. Ellis and A. Friedman, 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

V. Dolejší, M. Feistauer and J. Felcman, On the discrete Friedrichs inequality for non-conforming finite elements. Numer. Funct. Anal. Optim. 20 (1999) 437–447. | DOI | MR | Zbl

V. Dolejší, M. Feistauer, J. Felcman and A. Kliková, 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

A. Ern and J.-L. Guermond, Theory and Practice of Finite Elements. Appl. Math. Sci. 159, Springer, New York e.a. (2004). | MR | Zbl

R. Eymard, T. Gallouët, R. Herbin and J.-C. Latché, 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

R. Eymard, D. Hilhorst and M. Vohralik, A combined finite volume-nonconforming/mixed-hybrid finite element scheme for degenerate parabolic problems. Numer. Math. 105 (2006) 73–131. | DOI | MR | Zbl

R. Eymard, D. Hilhorst and M. Vohralik, 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

M. Feistauer, Mathematical Methods in Fluid Dynamics. Pitman Monographs and Surveys in Pure and Applied Mathematics 67. Longman, Harlow (1993). | MR | Zbl

M. Feistauer, J. Felcman and M. Lukáčová-Medvid’Ová, 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

M. Feistauer, J. Felcman, M. Lukáčová-Medvid’Ová and G. Warnecke, 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

M. Feistauer, J. Felcman and I. Straškraba, Mathematical and Computational Methods for Compressible Flow. Clarendon Press, Oxford (2003). | MR | Zbl

M. Feistauer, V. Kučera, K. Najzan and J. Prokopová, Analysis of a space-time discontinuous Galerkin method for nonlinear convection-diffusion problems. Numer. Math. 117 (2011) 251–288. | DOI | MR | Zbl

M. Feistauer, J. Slavík and P. Stupka, 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

S. Fučík, O. John and A. Kufner, Function Spaces. Noordhoff, Leyden (1977). | MR | Zbl

T. Gallouët, R. Herbin and J.-C. Latché, 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

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

L. Hallo, C. Le Ribault and M. Buffat, 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

P.-Wen Hsieh and S.-Y. Yang, On efficient least-squares finite element methods for convection-dominated problems. Comput. Methods Appl. Mech. Engrg. 199 (2009) 183–196. | DOI | MR | Zbl

A. Jonsson and H. Wallin, Function Spaces on Subsets of $R^{n}$ . 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

P. Knobloch, Uniform validity of discrete Friedrich’s inequality for general nonconforming finite element spaces. Numer. Funct. Anal. Optimiz. 22 (2001) 107–126. | DOI | MR | Zbl

R.C. Leal Toledo and V. Ruas, 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

Z. Li, Convergence analysis of an upwind mixed element method for advection diffusion problems. Appl. Math. Comput. 212 (2009) 318–326. | MR | Zbl

A.S. Mounim, A stabilized finite element method for convection-diffusion problems. Numer. Methods Partial Differ. Equ. 28 (2012) 1916–1943. | DOI | MR | Zbl

K. Ohmori and T. Ushijima, 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

A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations. Springer Series in Computational Mathematics 23. Springer, Berlin e.a. (1994). | MR | Zbl

Y. Ren, A. Cheng and H. Wang, 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

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

R.Verfürth, Robust a posteriori error estimates for steady convection-diffusion equations. SIAM J. Numer. Anal. 43 (2005) 1766–1782. | DOI | MR | Zbl

R.Verfürth, Robust a posteriori error estimates for unsteady convection-diffusion equations. SIAM J. Numer. Anal. 43 (2005) 1783–1802. | DOI | MR | Zbl

M. Vlasák, V. Dolejší and J. Hájek, 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

M. Vohralík, On the discrete Poincaré-Friedrichs inequalities for nonconforming approximations of the Sobolev space $H^{1}$ . 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

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