On the convergence and convergence order of finite volume gradient schemes for oblique derivative boundary value problem
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On the convergence and convergence order of finite volume gradient schemes for oblique derivative boundary value problems Abdallah Bradji1
· Jürgen Fuhrmann2
Received: 28 December 2016 / Revised: 17 May 2017 / Accepted: 1 June 2017 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2017
Abstract This work is an improvement of the previous note (Bradji in: Fuhrmann et al., Finite volumes for complex applications VII–methods, theoretical aspects. Proceedings of the FVCA 7, Berlin, 2014) which dealt with the convergence analysis of a finite volume scheme for the Poisson’s equation with a linear oblique derivative boundary condition. The formulation of the finite volume scheme given in Bradji (in: Fuhrmann et al., Finite volumes for complex applications VII–methods, theoretical aspects. Proceedings of the FVCA 7, Berlin, 2014) involves the discrete gradient introduced recently in Eymard et al. (IMA J Numer Anal 30(4):1009–1043, 2010). In this paper, we consider the convergence analysis of finite volume schemes involving the discrete gradient of Eymard et al. (IMA J Numer Anal 30(4):1009–1043, 2010) for elliptic and parabolic equations with linear oblique derivative boundary conditions. Linear oblique derivative boundary conditions arise for instance in the study of the motion of water in a canal, cf. Lesnic (Commun Numer Methods Eng 23(12):1071–1080, 2007). We derive error estimates in several norms which allow us to get error estimates for the approximations of the exact solutions and its first derivatives. In particular, we provide an error estimate between the gradient of the exact solutions and the discrete gradient of the approximate solutions. Convergence of the family of finite volume approximate solutions towards the exact solution under weak regularity assumption is also investigated. In the case of parabolic equations with oblique derivative boundary conditions, we develop a new discrete a priori estimate result. The proof of this result is based on the use of a discrete mean Poincaré–Wirtinger inequality. Thanks to the stated a priori estimate
Communicated by Raphaèle Herbin.
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Abdallah Bradji [email protected]; [email protected] https://www.i2m.univ-amu.fr/∼bradji/ Jürgen Fuhrmann [email protected] http://www.wias-berlin.de/∼fuhrmann
1
Department of Mathematics, University of Badji Mokhtar, Annaba, Algeria
2
Weierstrass Institute, Mohrenstrasse 39, 10117 Berlin, Germany
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A. Bradji, J. Fuhrmann
and to a comparison with an appropriately chosen auxiliary finite volume scheme, we derive the convergence results. This work can be viewed as a continuation of the previous work (Bradji and Gallouët in Int J Finite Vol 3(2):1–35, 2006) where a convergence analysis for a finite volume scheme, based on the admissible mesh of Eymard et al. (In: Ciarlet and Lions, Handbook of numerical analysis, North-Holland, Amsterdam, 2000), for the Poisson’s equation with a linear oblique derivative boundary conditions is given. The obtained convergence results d
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