# Adaptive asynchronous time-stepping, stopping criteria, and a posteriori error estimates for fixed-stress iterative schemes for coupled poromechanics problems

## TL;DR

This paper introduces adaptive asynchronous time-stepping and stopping criteria for fixed-stress iterative schemes in coupled poromechanics, utilizing a posteriori error estimates to improve accuracy and efficiency in simulations.

## Contribution

It develops a posteriori error estimates and adaptive algorithms for fixed-stress schemes, enabling more efficient and accurate coupled poromechanics simulations.

## Key findings

- Guaranteed and fully computable error bounds are derived.
- Adaptive algorithms improve convergence and computational efficiency.
- Numerical experiments confirm the effectiveness of the proposed methods.

## Abstract

In this paper we develop adaptive iterative coupling schemes for the Biot system modeling coupled poromechanics problems. We particularly consider the space-time formulation of the fixed-stress iterative scheme, in which we first solve the problem of flow over the whole space-time interval, then exploiting the space-time information for solving the mechanics. Two common discretizations of this algorithm are then introduced based on two coupled mixed finite element methods in-space and the backward Euler scheme in-time. Therefrom, adaptive fixed-stress algorithms are build on conforming reconstructions of the pressure and displacement together with equilibrated flux and stresses reconstructions. These ingredients are used to derive a posteriori error estimates for the fixed-stress algorithms, distinguishing the different error components, namely the spatial discretization, the temporal discretization, and the fixed-stress iteration components. Precisely, at the iteration $k\geq 1$ of the adaptive algorithm, we prove that our estimate gives a guaranteed and fully computable upper bound on the energy-type error measuring the difference between the exact and approximate pressure and displacement. These error components are efficiently used to design adaptive asynchronous time-stepping and adaptive stopping criteria for the fixed-stress algorithms. Numerical experiments illustrate the efficiency of our estimates and the performance of the adaptive iterative coupling algorithms.