A nonlinear time compactness result and applications to discretization of degenerate parabolic-elliptic PDEs
This provides a theoretical tool for analyzing convergence of numerical schemes for degenerate parabolic problems, addressing a known bottleneck in handling variable time steps and multistep methods.
The authors develop a new discrete functional analysis compactness result for degenerate parabolic-elliptic PDEs, enabling convergence proofs for fully discrete schemes with variable time steps and multistep methods like BDF2. They demonstrate its application by proving convergence of a finite volume/BDF2 scheme for the porous medium equation.
We propose a discrete functional analysis result suitable for proving compactness in the framework of fully discrete approximations of strongly degenerate parabolic problems. It is based on the original exploitation of a result related to compensated compactness rather than on a classical estimate on the space and time translates in the spirit of Simon (Ann. Mat. Pura Appl. 1987). Our approach allows to handle various numerical discretizations both in the space variables and in the time variable. In particular, we can cope quite easily with variable time steps and with multistep time differentiation methods like, e.g., the backward differentiation formula of order 2 (BDF2) scheme. We illustrate our approach by proving the convergence of a two-point flux Finite Volume in space and BDF2 in time approximation of the porous medium equation.