Time Compactness Tools for Discretized Evolution Equations and Applications to Degenerate Parabolic PDEs

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 Gallouet and Latche [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.

[1]  R. Eymard,et al.  Finite Volume Methods , 2019, Computational Methods for Fluid Dynamics.

[2]  Mostafa Bendahmane,et al.  Well-posedness results for triply nonlinear degenerate parabolic equations , 2008, 0810.2326.

[3]  B. Andreianov,et al.  Convergence of approximate solutions to an elliptic–parabolic equation without the structure condition , 2011, Nonlinear Differential Equations and Applications NoDEA.

[4]  B. Andreianov,et al.  Structural stability for variable exponent elliptic problems, I: The p(x)-Laplacian kind problems , 2010 .

[5]  Thierry Gallouët,et al.  Compactness of discrete approximate solutions to parabolic PDEs - Application to a turbulence model , 2012 .

[6]  Herbert Amann,et al.  Compact embeddings of vector valued Sobolev and Besov spaces , 2000 .

[7]  K. Karlsen,et al.  Discrete duality finite volume schemes for doubly nonlinear degenerate hyperbolic-parabolic equations , 2009, 0901.0816.

[8]  Ricardo Ruiz-Baier,et al.  Analysis of a finite volume method for a cross-diffusion model in population dynamics , 2011 .

[9]  Etienne Emmrich,et al.  Doubly nonlinear evolution equations of second order: Existence and fully discrete approximation , 2011 .

[10]  J. Simon.  Compact sets in the spaceLp(O,T; B) , 1986 .

[11]  Stephan Luckhaus,et al.  Quasilinear elliptic-parabolic differential equations , 1983 .

[12]  S. N. Kruzhkov,et al.  Results concerning the nature of the continuity of solutions of parabolic equations and some of their applications , 1969 .

[13]  Mostafa Bendahmane,et al.  On 3D DDFV Discretization of Gradient and Divergence Operators: Discrete Functional Analysis Tools and Applications to Degenerate Parabolic Problems , 2013, Comput. Methods Appl. Math..

[14]  Thierry Gallouët,et al.  Convergence of a finite volume scheme for nonlinear degenerate parabolic equations , 2002, Numerische Mathematik.

[15]  Kenneth H. Karlsen,et al.  Convergence of discrete duality finite volume schemes for the cardiac bidomain model , 2010, Networks Heterog. Media.