Guaranteed and computable error bounds for approximations constructed by an iterative decoupling of the Biot problem

TL;DR

This paper develops guaranteed, fully computable a posteriori error bounds for iterative approximations of the Biot problem, enabling reliable error estimation and adaptive refinement in poroelastic media simulations.

Contribution

It introduces a novel approach combining contractive mapping estimates with elliptic PDE functionals for error bounds in iterative Biot problem solutions.

Findings

Error bounds are independent of discretization method.

Bounds are fully computable and mesh-independent.

Error indicators support adaptive procedures.

Abstract

The paper is concerned with guaranteed a posteriori error estimates for a class of evolutionary problems related to poroelastic media governed by the quasi-static linear Biot equations. The system is decoupled employing the fixed-stress split scheme, which leads to a semi-discrete system solved iteratively. The error bounds are derived by combining a posteriori estimates for contractive mappings with those of the functional type for elliptic partial differential equations. The estimates are applicable for any approximation in the admissible functional space and are independent of the discretization method. They are fully computable, do not contain mesh dependent constants, and provide reliable global estimates of the error measured in the energy norm. Moreover, they suggest efficient error indicators for the distribution of local errors, which can be used in adaptive procedures.