A nonlinear time compactness result and applications to discretization of degenerate parabolic-elliptic PDEs

TL;DR

This paper introduces a novel discrete compactness result for fully discrete approximations of degenerate parabolic PDEs, enabling convergence proofs for various numerical schemes including variable time steps and multistep methods.

Contribution

It develops a new compactness framework based on compensated compactness, applicable to diverse discretizations and time-stepping schemes for degenerate parabolic equations.

Findings

Proves convergence of a finite volume and BDF2 scheme for the porous medium equation.

Handles variable time steps and multistep methods effectively.

Provides a unified approach for discrete compactness in degenerate PDE discretizations.

Abstract

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.