A Posteriori Error Estimates for Approximate Solutions of the Barenblatt-Biot Poroelastic Model

Abstract

Abstract. The paper is concerned with the Barenblatt-Biott model in the theory of poroelasticity. We derive a guaranteed estimate of the difference between exact and approximate solutions expressed in a combined norm that encompasses errors for the pressure fields computed from the diffusion part of the model and errors related to stresses (strains) of the elastic part. Estimates do not contain generic (mesh-dependent) constants and are valid for any conforming approximation of pressure and stress fields.

A Mixed Finite Element Method for Nearly Incompressible Multiple-Network Poroelasticity

“…[32,25,29,2,28,22,38]; to the best of our knowledge, the general multiple-network poroelasticity equations have received much less attention, especially from the numerical perspective. The case A = 2 is known as the Barenblatt-Biot model, and we note that Showalter and Momken [33] present an existence analysis for this model, while Nordbotten and co-authors [27] present an a posteriori error analysis for an approximation of a static Barenblatt-Biot system.…”

A-posteriori error estimation and adaptivity for multiple-network poroelasticity

“…The a-posteriori landscape for the more general MPET system (1.1) is considerably sparse. A-posteriori error estimates for the two-field formulation of the Barenblatt-Biot equations (corresponding to the J = 2 case of (1.1)) have indeed been obtained by Nordbotten et al [2010]. In general, though, there has been little work on the development of a-posteriori error estimators for (1.1), for formulations with any number of fields, in the case of more than one fluid network (i.e.…”

Algebraic preconditioning for Biot-Barenblatt poroelastic systems

“…In this paper, the Biot-Barenblatt model will be described by five variablesdisplacement u, two fluxes v 1 , v 2 and two corresponding pressures p 1 , p 2 describing the fluid flow in matrix and fractures, respectively. The equations describing the model are [19]…”

“…Solvability, stability and other properties of the model and properties of the proposed discretization are not analysed in this paper, for this we can refer e.g. [14], [15], [16], [19], [20]. Instead, we consider systems to be solved in each time step of the implicit Euler method and propose and analyse block diagonal preconditioners, which generalize the preconditioner considered in [3], [4].…”