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Finite element error estimates for nonlinear convective problems

Václav Kučera


This paper is concerned with the analysis of the finite element method applied to a nonstationary nonlinear convective problem. Using special estimates of the convective terms, we prove a priori error estimates for an explicit, semidiscrete and implicit scheme. While the explicit case is rather straightforward via mathematical induction, for the semidiscrete scheme we need to apply so-called continuous mathematical induction and a nonlinear Gronwall lemma. For the implicit scheme, we use a suitable continuation of the discrete implicit solution and again use continuous mathematical induction to prove the error estimates. Finally, we extend the presented analysis from globally Lipschitz-continuous convective nonlinearities to the locally Lipschitz-continuous case.

MSC 2010: 65M15; 65M60; 65M12


This work is a part of the research projects P201/11/P414 and P201/13/00522S of the Czech Science Foundation. V. Kucera is a junior researcher at the University Center for Mathematical Modelling, Applied Analysis and Computational Mathematics (Math MAC).


[1] R. Bank, J. B. Burger, W. Fichtner, and R. Smith, Some up-winding techniques for finite element approximations of convection diffusion equation, Numer. Math. 58 (1990), 185–202. Search in Google Scholar

[2] Y. R. Chao, A note on ‘Continuous mathematical induction’, Bull. Amer. Math. Soc. 26 (1919), 17–18. Search in Google Scholar

[3] P.G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1979. Search in Google Scholar

[4] P. L. Clark, Real induction, Search in Google Scholar

[5] M. Dobrowolski and H.-G. Roos, A priori estimates for the solution of convection–diffusion problems and interpolation on Shishkin meshes, Z. Anal. Anwend., 16 (1997), 1001–1012. Search in Google Scholar

[6] V. Dolejší, M. Feistauer, and J. Hozman, Analysis of semi-implicit DGFEM for nonlinear convection–diffusion problems on nonconforming meshes, Comput. Meth. Appl. Mech. Engrg., 197 (2007), 2813–2827. Search in Google Scholar

[7] M. Feistauer, J. Felcman, and I. Straškraba, Mathematical and Computational Methods for Compressible Flow, Clarendon Press, Oxford, 2003. Search in Google Scholar

[8] V. Girault and P. A. Raviart, Finite Element Methods for Navier Stokes Equations, Theorems and Algorithms, Springer Verlag, 1986. Search in Google Scholar

[9] T. J. R. Hughes and A. Brooks, A multi-dimension upwind scheme with no crosswind diffusion, in: AMD 34 (1979), Am. Soc. Mech. Engrg., New York, 19–35. Search in Google Scholar

[10] J. Johnson, Numerical Solution of Partial Differential Equations by the Finite Element Method, Cambridge University Press, Cambridge, 1987. Search in Google Scholar

[11] V. Kučera, On diffusion-uniform error estimates for the DG method applied to singularly perturbed problems, IMA J. Numer. Anal. 34 (2014), 820–861. Search in Google Scholar

[12] A. Kufner, O. John and S. Fučík, Function Spaces, Academia, Prague, 1977. Search in Google Scholar

[13] X.G. Li, C.K. Chan, and S. Wang, The finite element method with weighted basis functions for singularly perturbed convection–diffusion problems, J. Comput. Phys., 195 (2004), 773–789. Search in Google Scholar

[14] H.-G. Roos, M. Stynes, and L. Tobiska, Robust Numerical Methods for Singularly Perturbed Differential Equations, Springer-Verlag, Berlin–Heidelberg, 2008. Search in Google Scholar

[15] E. Zeidler, Nonlinear Functional Analysis and Its Applications II/B: Nonlinear Monotone Operators, Springer, 1986. Search in Google Scholar

[16] Q. Zhang and C.-W. Shu, Error estimates to smooth solutions of Runge–Kutta discontinuous Galerkin methods for scalar conservation laws, SIAMJ. Numer. Anal., 42 (2004), 641–666. Search in Google Scholar

Received: 2015-3-9
Revised: 2015-9-14
Accepted: 2015-10-9
Published Online: 2016-10-5
Published in Print: 2016-10-1

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