A posteriori error estimates for the Richards equation

TL;DR

This paper develops reliable and efficient a posteriori error estimates for numerical solutions of the complex, degenerate Richards equation, enabling improved adaptive methods for modeling water flow in porous media.

Contribution

It introduces fully computable, locally space-time efficient error bounds for the Richards equation, including nonlocal-in-time estimates and error contributor analysis, applicable to nonlinear and degenerate cases.

Findings

Error estimators accurately identify errors up to a factor of unity.

Estimates are valid for both nondegenerate and degenerate cases.

Numerical tests confirm the effectiveness of the estimators.

Abstract

The Richards equation is commonly used to model the flow of water and air through soil, and it serves as a gateway equation for multiphase flows through porous media. It is a nonlinear advection-reaction-diffusion equation that exhibits both parabolic-hyperbolic and parabolic-elliptic kinds of degeneracies. In this study, we provide reliable, fully computable, and locally space-time efficient a posteriori error bounds for numerical approximations of the fully degenerate Richards equation. For showing global reliability, a nonlocal-in-time error estimate is derived individually for the time-integrated $H^1(H^{-1})$, $L^2(L^2)$, and the $L^2(H^1)$ errors. A maximum principle and a degeneracy estimator are employed for the last one. Global and local space-time efficiency error bounds are then obtained in a standard $H^1(H^{-1})\cap L^2(H^1)$ norm. The reliability and efficiency norms employed coincide when there is no nonlinearity. Moreover, error contributors such as flux nonconformity, time discretization, quadrature, linearization, and data oscillation are identified and separated. The estimates are also valid in a setting where iterative linearization with inexact solvers is considered. Numerical tests are conducted for nondegenerate and degenerate cases having exact solutions, as well as for a realistic case. It is shown that the estimators correctly identify the errors up to a factor of the order of unity.