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A nonlinear time compactness result and applications to discretization of degenerate parabolic-elliptic PDEs. (English) Zbl 1387.46051
In the discretization of evolution PDEs, the proofs of convergence of numerical schemes need sufficiently strong compactness arguments in passing to the limit. In this paper, the authors propose a discrete functional analysis result suitable for proving compactness in the framework of fully discrete approximations of strongly degenerate parabolic problems. The approach is based on a deep result related to compensated compactness. This approach is very powerful because it can be applied to various numerical discretizations both in the space variables and in the time variable. This approach works quite well with variable time steps and with multistep time differentiation methods such as the backward differentiation formula of order 2 (BDF2) scheme. For this reason, the authors give an application of their method in the proof of the convergence of a two-point flux finite volume in space and BDF2 in time approximation of the porous medium equation.

MSC:
46N40 Applications of functional analysis in numerical analysis
46N20 Applications of functional analysis to differential and integral equations
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35K65 Degenerate parabolic equations
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