Robust a posteriori error analysis for rotation-based formulations of the elasticity/poroelasticity coupling

TL;DR

This paper develops robust residual-based a posteriori error estimators for rotation-based mixed finite element methods in elasticity and poroelasticity, valid in 2D and 3D, with demonstrated numerical effectiveness.

Contribution

It introduces new residual-based a posteriori error estimators for rotation-based formulations, with rigorous bounds and robustness across parameters and dimensions.

Findings

Estimators provide reliable upper and lower bounds for errors.

Methods are robust with respect to model parameters and applicable in 2D and 3D.

Numerical experiments confirm the effectiveness of adaptive mesh refinement.

Abstract

We develop the \textit{a posteriori} error analysis of three mixed finite element formulations for rotation-based equations in elasticity, poroelasticity, and interfacial elasticity-poroelasticity. The discretisations use $H^1$-conforming finite elements of degree $k+1$ for displacement and fluid pressure, and discontinuous piecewise polynomials of degree $k$ for rotation vector, total pressure, and elastic pressure. Residual-based estimators are constructed, and upper and lower bounds (up to data oscillations) for all global estimators are rigorously derived. The methods are all robust with respect to the model parameters (in particular, the Lam\'e constants), they are valid in 2D and 3D, and also for arbitrary polynomial degree $k\geq 0$. The error behaviour predicted by the theoretical analysis is then demonstrated numerically on a set of computational examples including different geometries on which we perform adaptive mesh refinement guided by the \textit{a posteriori} error estimators.