Accessible Requires Authentication Published by De Gruyter October 5, 2016

Finite element error estimates for nonlinear convective problems

Václav Kučera

Abstract

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

Funding

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).

References

[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, http://citeseer.ist.psu.edu/viewdoc/summary?doi=10.1.1.187.3514. 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

© 2016 Walter de Gruyter GmbH, Berlin/Boston