An initial-boundary value problem describing moisture transport in porous media: existence of strong solutions and an error estimate for a finite volume scheme

TL;DR

This work analyzes a Richards-type moisture-transport model in porous media, formulating an initial-boundary value problem with space-time dependent data. It develops a finite-volume method (FVM) and rigorously proves the existence of a strong solution $v$ in the framework $v \\in W^{1,2}(0,T; H) \\cap L^ obreakfty(0,T; X) \\cap L^2(0,T; H^2(0,1))$, even when the forcing $P$ depends on $(t,x)$. An explicit error estimate between the strong solution and the FVM approximation $v^{(n)}$ is derived, showing $|v - v^{(n)}|_{L^ obreakfty(0,T; H)}^2 + abla$-term bound $\\le C (\\Delta x^{(n)})^{1/2}$, with a crucial Gagliardo–Nirenberg inequality for piecewise-constant comparisons driving the analysis. The results establish convergence of the FVM and quantify its accuracy, supporting reliable simulations of moisture transport in porous media with time-varying boundary data or sources.

Abstract

We consider an initial-boundary value problem motivated by a mathematical model of moisture transport in porous media. We establish the existence of strong solutions and provide an error estimate for the approximate solutions constructed by the finite volume method. In the proof of the error estimate, the Gagliardo--Nirenberg type inequality for the difference between a continuous function and a piecewise constant function plays an important role.

Theorems & Definitions (18)

• Theorem 2.1
• Remark 2.2
• Theorem 2.3
• Lemma 2.4
• Definition 3.1
• Proposition 3.2: M-A-1
• Lemma 3.3: M-A-1
• Remark 3.4
• Lemma 3.5
• proof