Mathematics of Computation

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ISSN 1088-6842(e) ISSN 0025-5718(p)

Convergence of a finite volume scheme for the convection-diffusion equation with $\mathrm{L}^1$ data

Authors: T. Gallouët, A. Larcher and J. C. Latché
Journal: Math. Comp. 81 (2012), 1429-1454
MSC (2010): Primary 35K10, 65M12, 65M08
Posted: December 21, 2011
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Abstract: In this paper, we prove the convergence of a finite-volume scheme for the time-dependent convection-diffusion equation with an $\mathrm {L}^1$ right-hand side. To this purpose, we first prove estimates for the discrete solution and for its discrete time and space derivatives. Then we show the convergence of a sequence of discrete solutions obtained with more and more refined discretizations, possibly up to the extraction of a subsequence, to a function which meets the regularity requirements of the weak formulation of the problem; to this purpose, we prove a compactness result, which may be seen as a discrete analogue to the Aubin-Simon lemma. Finally, such a limit is shown to be indeed a weak solution.

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