A posteriori error estimates for vertex centered finite volume approximations of convection-diffusion-reaction equations

Ohlberger, Mario

ESAIM: Modélisation mathématique et analyse numérique, Tome 35 (2001) no. 2, pp. 355-387.

This paper is devoted to the study of a posteriori error estimates for the scalar nonlinear convection-diffusion-reaction equation $c_t+\nabla ·\left(𝐮f\left(c\right)\right)-\nabla ·\left(D\nabla c\right)+\lambda c=0$. The estimates for the error between the exact solution and an upwind finite volume approximation to the solution are derived in the $L^1$-norm, independent of the diffusion parameter $D$. The resulting a posteriori error estimate is used to define an grid adaptive solution algorithm for the finite volume scheme. Finally numerical experiments underline the applicability of the theoretical results.

[1] L. Angermann, An introduction to finite volume methods for linear elliptic equations of second order. Preprint 164, Institut für Angewandte Mathematik, Universität Erlangen (1995). | MR

[2] Lutz Angermann, A finite element method for the numerical solution of convection-dominated anisotropic diffusion equations. Numer. Math. 85 (2000) 175-195. | Zbl

[3] P. Angot, V. Dolejší, M. Feistauer and J. Felcman, Analysis of a combined barycentric finite volume - finite element method for nonlinear convection diffusion problems. Appl. Math., Praha 43 (1998) 263-311. | Zbl

[4] I. Babuška and W.C. Rheinboldt, Error estimators for adaptive finite element computations. SIAM J. Numer. Anal. 15 (1978) 736-754. | Zbl

[5] E. Bänsch, Local mesh refinement in 2 and 3 dimensions. IMPACT Comput. Sci. Engrg. 3 (1991) 181-191. | Zbl

[6] R. Becker and R. Rannacher, A feed-back approach to error control in finite element methods: Basic analysis and examples. East-West J. Numer. Math. 4 (1996) 237-264. | Zbl

[7] F. Bouchut and B. Perthame, Kruzkov's estimates for scalar conservation laws revisited. Trans. Amer. Math. Soc. 350 (1998) 2847-2870. | Zbl

[8] J. Carrillo, Entropy solutions for nonlinear degenerate problems. Arch. Ration. Mech. Anal. 147 (1999) 269-361. | Zbl

[9] C. Chainais-Hillairet, Finite volume schemes for a nonlinear hyperbolic equation. Convergence towards the entropy solution and error estimates. ESAIM: M2AN 33 (1999) 129-156. | Numdam | Zbl

[10] S. Champier, Error estimates for the approximate solution of a nonlinear hyperbolic equation with source term given by finite volume scheme. Preprint, UMR 5585, Saint-Étienne University (1998). | MR

[11] G. Chavent and J. Jaffre, Mathematical models and finite elements for reservoir simulation. Elsevier, New York (1986). | Zbl

[12] B. Cockburn, F. Coquel and P.G. Lefloch, An error estimate for finite volume methods for multidimensional conservation laws. Math. Comput. 63 (1994) 77-103. | Zbl

[13] B. Cockburn and H. Gau, A posteriori error estimates for general numerical methods for scalar conservation laws. Comput. Appl. Math. 14 (1995) 37-47. | Zbl

[14] B. Cockburn and P.A. Gremaud, A priori error estimates for numerical methods for scalar conservation laws. Part I: The general approach. Math. Comput. 65 (1996) 533-573. | Zbl

[15] B. Cockburn and G. Gripenberg, Continuous dependence on the nonlinearities of solutions of degenerate parabolic equations. J. Differential Equations 151 (1999) 231-251. | Zbl

[16] W. Dörfler, Uniformly convergent finite-element methods for singularly perturbed convection-diffusion equations. Habilitationsschrift, Mathematische Fakultät, Freiburg (1998).

[17] K. Eriksson and C. Johnson, Adaptive streamline diffusion finite element methods for stationary convection-diffusion problems. Math. Comput. 60 (1993) 167-188. | Zbl

[18] K. Eriksson and C. Johnson, Adaptive finite element methods for parabolic problems 32 (1995) 706-740. | Zbl

[19] K. Eriksson and C. Johnson, Adaptive finite element methods for parabolic problems. IV: Nonlinear Problems. SIAM J. Numer. Anal. 32 (1995) 1729-1749. | Zbl

[20] S. Evje, K.H. Karlsen and N.H. Risebro, A continuous dependence result for nonlinear degenerate parabolic equations with spatial dependent flux function. Preprint, Department of Mathematics, Bergen University (2000). | MR

[21] R. Eymard, T. Gallouët, M. Ghilani and R. Herbin, Error estimates for the approximate solution of a nonlinear hyperbolic equation given by finite volume schemes. IMA J. Numer. Anal. 18 (1998) 563-594. | Zbl

[22] R. Eymard, T. Gallouët, R. Herbin and A. Michel, Convergence of a finite volume scheme for nonlinear degenerate parabolic equations. Preprint LATP 00-20, CMI, Provence University, Marseille (2000). | MR

[23] P. Frolkovic, Maximum principle and local mass balance for numerical solutions of transport equations coupled with variable density flow. Acta Math. Univ. Comenian. 67 (1998) 137-157. | Zbl

[24] J. Fuhrmann and H. Langmach, Stability and existence of solutions of time-implicit finite volume schemes for viscous nonlinear conservation laws. Preprint 437, Weierstraß-Institut, Berlin (1998). | MR

[25] R. Helmig, Multiphase flow and transport processes in the subsurface: A contribution to the modeling of hydrosystems. Springer, Berlin, Heidelberg (1997).

[26] R. Herbin, An error estimate for a finite volume scheme for a diffusion-convection problem on a triangular mesh. Numer. Methods Partial Differential Equation 11 (1995) 165-173. | Zbl

[27] P. Houston and E. Süli, Adaptive lagrange-galerkin methods for unsteady convection-dominated diffusion problems. Report 95/24, Numerical Analysis Group, Oxford University Computing Laboratory (1995).

[28] J. Jaffre, Décentrage et élements finis mixtes pour les équations de diffusion-convection. Calcolo 21 (1984) 171-197. | Zbl

[29] V. John, J.M. Maubach and L. Tobiska, Nonconforming streamline-diffusion-finite-element-methods for convection-diffusion problems. Numer. Math. 78 (1997) 165-188. | Zbl

[30] C. Johnson, Finite element methods for convection-diffusion problems, in Proc. 5th Int. Symp. (Versailles, 1981), Computing methods in applied sciences and engineering V (1982) 311-323. | Zbl

[31] K.H. Karlsen and N.H. Risebro, On the uniqueness and stability of entropy solutions of nonlinear degenerate parabolic equations with rough coefficients. Preprint 143, Department of Mathematics, Bergen University (2000). | MR

[32] D. Kröner, Numerical schemes for conservation laws. Teubner, Stuttgart (1997). | MR | Zbl

[33] D. Kröner and M. Ohlberger, A posteriori error estimates for upwind finite volume schemes for nonlinear conservation laws in multi dimensions. Math. Comput. 69 (2000) 25-39. | Zbl

[34] D. Kröner and M. Rokyta, A priori error estimates for upwind finite volume schemes in several space dimensions. Preprint 37, Math. Fakultät, Freiburg (1996).

[35] S.N. Kruzkov, First order quasilinear equations in several independent variables. Math. USSR Sbornik 10 (1970) 217-243. | Zbl

[36] N.N. Kuznetsov, Accuracy of some approximate methods for computing the weak solutions of a first-order quasi-linear equation. USSR, Comput. Math. Math. Phys. 16 (1976) 159-193. | Zbl

[37] J. Málek, J. Nečas, M. Rokyta and M. Růžička, Weak and measure-valued solutions to evolutionary PDEs, in Applied Mathematics and Mathematical Computation 13, Chapman and Hall, London, Weinheim, New York, Tokyo, Melbourne, Madras (1968). | MR | Zbl

[38] M. Marion and A. Mollard, An adaptive multi-level method for convection diffusion problems. ESAIM: M2AN 34 (2000) 439-458. | Numdam | Zbl

[39] R.H. Nochetto, A. Schmidt and C. Verdi, A posteriori error estimation and adaptivity for degenerate parabolic problems. Math. Comput. 69 (2000) 1-24. | Zbl

[40] M. Ohlberger, Convergence of a mixed finite element-finite volume method for the two phase flow in porous media. East-West J. Numer. Math. 5 (1997) 183-210. | Zbl

[41] M. Ohlberger, A posteriori error estimates for finite volume approximations to singularly perturbed nonlinear convection-diffusion equations. Numer. Math. 87 (2001) 737-761. | Zbl

[42] Ch. Rohde, Entropy solutions for weakly coupled hyperbolic systems in several space dimensions. Z. Angew. Math. Phys. 49 (1998) 470-499. | Zbl

[43] Ch. Rohde, Upwind finite volume schemes for weakly coupled hyperbolic systems of conservation laws in 2D. Numer. Math. 81 (1998) 85-123. | Zbl

[44] H.-G. Roos, M. Stynes and L. Tobiska, Numerical methods for singularly perturbed differential equations. Convection-diffusion and flow problems, in Springer Ser. Comput. Math. 24, Springer-Verlag, Berlin (1996). | MR

[45] E. Tadmor, Local error estimates for discontinuous solutions of nonlinear hyperbolic equations. SIAM J. Numer. Anal. 28 (1991) 891-906. | Zbl

[46] R. Verfürth, A review of a posteriori error estimation and adaptive mesh-refinement techniques, in Wiley-Teubner Ser. Adv. Numer. Math., Teubner, Stuttgart (1996). | Zbl

[47] R. Verfürth, A posteriori error estimators for convection-diffusion equations. Numer. Math. 80 (1998) 641-663. | Zbl

[48] J.P. Vila, Convergence and error estimates in finite volume schemes for general multi-dimensional scalar conservation laws. I Explicit monotone schemes. ESAIM: M2AN 28 (1994) 267-295. | Numdam | Zbl