Mathematical analysis for a doubly degenerate parabolic equation: Application to the Richards equation

TL;DR

This paper provides a rigorous mathematical analysis of a doubly degenerate parabolic equation, specifically applied to the Richards equation, establishing existence, convergence, and physical bounds preservation without strict positivity assumptions.

Contribution

It introduces a novel analysis framework using a bounded auxiliary variable and weighted Sobolev spaces for the Richards equation, ensuring convergence and physical bounds preservation.

Findings

Proved existence of weak solutions without positive lower bounds on diffusivity.

Established unconditional linear convergence of the L-scheme linearization.

Demonstrated preservation of physical saturation bounds.

Abstract

This paper presents a mathematical analysis of a doubly degenerate parabolic equation and its application to the Richards equation using a bounded auxiliary variable. We establish the existence of weak solutions using semi-implicit time discretization combined with maximal monotone operator theory. The analysis is conducted within weighted Sobolev spaces, allowing for a rigorous treatment of the equation's strict degeneracy and strong nonlinearities. A key feature of this study is the derivation of convergence results without imposing strictly positive lower bounds on the diffusivity or requiring high regularity of the solution. Furthermore, we prove that the Richards equation using the introduced auxiliary variable preserves the physical bounds of the saturation and demonstrate the unconditional linear convergence of the L-scheme linearization to the semi-discrete solution.