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.
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MSC2010: Primary: 65M12, 46B50, 35K65; Secondary: 46E39, 65M08
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References
H.W. Alt and S. Luckhaus. Quasilinear elliptic-parabolic differential equations. Mat. Z., (1983), 183:311–341.
H. Amann. Compact embeddings of vector-valued Sobolev and Besov spaces. Glasnik Matematički, (2000), 35:161–177.
B. Andreianov, M. Bendahmane and F. Hubert. On 3D DDFV discretization of gradient and divergence operators. II. Discrete functional analysis tools and applications to degenerate parabolic problems. Preprint HAL, (2011), http://hal.archives-ouvertes.fr/hal-00567342
B. Andreianov, M. Bendahmane, and K.H. Karlsen. Discrete duality finite volume schemes for doubly nonlinear degenerate hyperbolic-parabolic equations. J. Hyp. Diff. Eq., (2010), 7:1–67.
B. Andreianov, M. Bendahmane, K.H. Karlsen and S. Ouaro. Well-posedness results for triply nonlinear degenerate parabolic equations. J. Diff. Eq., (2009), 247(1):277–302.
B. Andreianov, M. Bendahmane, K.H. Karlsen and Ch. Pierre. Convergence of Discrete Duality Finite Volume schemes for the macroscopic bidomain model of the heart electric activity. Netw. Het. Media, (2011), to appear; available at http://hal.archives-ouvertes.fr/hal-00526047
B. Andreianov, M. Bendahmane and R. Ruiz Baier. Analysis of a finite volume method to solve a cross-diffusion population system. Math. Models Meth. Appl. Sci., (2011), to appear.
B. Andreianov and P. Wittbold. Convergence of approximate solutions to an elliptic-parabolic equation without the structure condition. Preprint, (2011).
J.-P. Aubin. Un théorème de compacité. (French) C.R. Acad. Sc. Paris, (1963), 256:5042–5044.
Ph. Bénilan and P. Wittbold. Sur un problème parabolique-elliptique. (French) M2AN Math. Modelling and Num. Anal., (1999), 33(1):121–127.
J.A. Dubinskii. Weak convergence for elliptic and parabolic equations. (Russian) Math. USSR Sbornik, (1965), 67:609–642.
E. Emmrich and M. Thalhammer. Doubly nonlinear evolution equations of second order: Existence and fully discrete approximation. J. Diff. Eq., (2011), to appear.
R. Eymard, T. Gallouët, and R. Herbin. Finite Volume Methods. Handbook of Numerical Analysis, Vol. VII (2000). P. Ciarlet, J.-L. Lions, eds., North-Holland.
R. Eymard, T. Gallouët, R. Herbin and A. Michel. Convergence of a finite volume scheme for nonlinear degenerate parabolic equations. Numer. Math., (2002), 92(1):41–82.
T. Gallouët and J.-C. Latché. Compactness of discrete approximate solutions to parabolic PDEs - Application to a turbulence model. Comm. on Pure and Appl. Anal., (2011), to appear.
S.N. Kruzhkov. Results on the nature of the continuity of solutions of parabolic equations and some of their applications. Mat. Zametki (Math. Notes), (1969), 6(1):517-523.
J. Simon. Compact sets in the space L p(0, T; B). Ann. Mat. Pura ed Appl., (1987), 146:65–96.
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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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