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

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.

Figures (3)

• Figure 1: Extension of the saturation function $\theta(\eta)$ for parameters $c=\frac{5}{3}$ and $b = \frac{3}{5}$.
• Figure 2: Extension of the diffusivity function $K(\eta)$ with parameters $K_s C = 1$, $m=0.6$, and $a=\frac{5}{3}$.
• Figure 3: Comparison of the extended functions $\bar{K}_z$ and $\bar{K}_{1z}$.

Theorems & Definitions (36)

• Definition 2.1
• Remark 2.1
• Theorem 2.1
• Proposition 4.1
• proof
• Theorem 4.1
• Definition 4.1: Proposition 2.2 in brezis1973ope
• Lemma 4.1: Proposition 2.5 in brezis1973ope
• Lemma 4.2
• proof