Discrete positivity and maximum principles for a finite element discretization of the Richards equation

TL;DR

This paper introduces a semi-implicit finite element method for the Richards equation that guarantees physical bounds like positivity and maximum principles, even in degenerate and dry soil conditions.

Contribution

It develops a novel bounded auxiliary variable framework with rigorous conditions ensuring discrete positivity and maximum principles, improving stability and physical fidelity.

Findings

Ensures solutions respect physical bounds across various flow regimes.

Provides mathematical proofs for geometric and algebraic constraints.

Demonstrates effectiveness through comprehensive numerical validation.

Abstract

Standard finite element discretizations of the Richards equation may violate the discrete minimum principle, producing unphysical negative saturations. While existing bound-preserving methods typically rely on computationally expensive fully implicit solvers, we propose a novel semi-implicit finite element framework utilizing a bounded continuous auxiliary variable. Our approach treats the gravity-driven advective term using a linearly implicit technique, which improves the time-step restrictions required by explicit gravity methods near the degenerate limit. We provide rigorous mathematical proofs establishing sufficient geometric and algebraic constraints for discrete positivity and the discrete maximum principle, specifically a local P\'eclet condition and a discrete row-sum condition. When both conditions are satisfied on weakly acute meshes with mass lumping, our framework ensures that numerical solutions strictly respect physical bounds across highly degenerate conditions and initially dry soil regimes. Comprehensive numerical validation demonstrates the method across multiple flow regimes, including cases where algebraic conditions are satisfied, violated, and recovered through mesh refinement.