Abstract
We discuss several techniques for proving compactness of sequences of approximate solutions to discretized evolution PDEs. While the well-known Aubin-Simon kind functional-analytic techniques were recently generalized to the discrete setting by Gallouët and Latché [15], here we discuss direct techniques for estimating the time translates of approximate solutions in the space L 1. One important result is the Kruzhkov time compactness lemma. Further, we describe a specific technique that relies upon the order-preservation property. Motivation comes from studying convergence of finite volume discretizations for various classes of nonlinear degenerate parabolic equations. These and other applications are briefly described.
MSC2010: Primary: 65M12, 46B50, 35K65; Secondary: 46E39, 65M08
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Acknowledgements
The author thanks E. Emmrich for discussions on the above techniques.
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Andreianov, B. (2011). Time Compactness Tools for Discretized Evolution Equations and Applications to Degenerate Parabolic PDEs. In: Fořt, J., Fürst, J., Halama, J., Herbin, R., Hubert, F. (eds) Finite Volumes for Complex Applications VI Problems & Perspectives. Springer Proceedings in Mathematics, vol 4. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-20671-9_3
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DOI: https://doi.org/10.1007/978-3-642-20671-9_3
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