Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Elyes Ahmed, Jan Martin Nordbotten, Florin Adrian Radu

TL;DR

This work advances adaptive fixed-stress coupling for Biot's poroelasticity by deriving guaranteed, computable a posteriori energy-type error estimates that decompose the total error into spatial, temporal, and fixed-stress iterative components. It develops space-time schemes with two standard MFE discretizations and two post-processing steps, then introduces adaptive asynchronous time stepping and adaptive stopping criteria guided by robust estimators. The framework relies on $H^{1}$- and $\mathbf{H}(\textnormal{div})$-conforming reconstructions and equilibrated flux and stress reconstructions to obtain reliable error bounds that drive adaptivity. Numerical experiments demonstrate substantial reductions in fixed-stress iterations and overall computational cost while preserving accuracy, with potential extensions to other coupling strategies and conforming discretizations.

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.

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

TL;DR

This work advances adaptive fixed-stress coupling for Biot's poroelasticity by deriving guaranteed, computable a posteriori energy-type error estimates that decompose the total error into spatial, temporal, and fixed-stress iterative components. It develops space-time schemes with two standard MFE discretizations and two post-processing steps, then introduces adaptive asynchronous time stepping and adaptive stopping criteria guided by robust estimators. The framework relies on - and -conforming reconstructions and equilibrated flux and stress reconstructions to obtain reliable error bounds that drive adaptivity. Numerical experiments demonstrate substantial reductions in fixed-stress iterations and overall computational cost while preserving accuracy, with potential extensions to other coupling strategies and conforming discretizations.

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 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.

Paper Structure

This paper contains 25 sections, 3 theorems, 68 equations, 10 figures, 1 table.

Table of Contents

  1. Introduction
  2. Notation
  3. Partitions of the time interval $(0,T)$
  4. Partition of the domain $\Omega$
  5. Discrete function spaces
  6. Fully discrete space-time fixed-stress schemes based on MFE in-space and the backward Euler scheme in-time
  7. Two standard discrete fixed-stress schemes.
  8. Post-processing
  9. The adaptive fixed-stress algorithms
  10. Methodology for adaptive asynchronous time-stepping and adaptive stopping criteria
  11. The adaptive algorithms
  12. Concept of $H^1$-, and $\mathbf{H}(\textnormal{div})$-conforming reconstructions in Biot's poro-elasticity system
  13. Pressure and displacement reconstructions
  14. Equilibrated flux $\hat{{\mathbf{w}}}^{k}_{h\tau}$ and stress $\hat{\boldsymbol{\sigma}}^{k}_{h\tau}$ reconstructions
  15. The a posteriori error estimates
  16. ...and 10 more sections

Key Result

Lemma 5.1

At each iteration $k\geq 1$ of Algorithm space_time_discrete_adaptive, let $(\widetilde{p}^{k}_{h\tau},\widetilde{{\mathbf{u}}}^{k}_{h\tau})$ be the post-processed pressure and displacement, and $(\hat{p}^{k}_{h\tau},\hat{{\mathbf{u}}}^{k}_{h\tau})$ be the reconstructed pressure and displacement. Th

Figures (10)

  • Figure 1: Number of fixed-stress iterations.
  • Figure 2: Spatial distributions of the discretization and fixed-stress errors and of the corresponding estimates at $t=T$.
  • Figure 3: Effectivity index
  • Figure 4: The standard and adpative multi-rate fixed-stress for various ratios of discretization in time $\delta_{\textnormal{fm}}=8,\,4,\,2$.
  • Figure 5: The ratio of the time error over the space error for the flow (top left) and mechanics (top right) problems as a function of the total number of space--time unknowns. Comparison of the induced overall errors (bottom left). The total and fixed-stress estimators as a function of fixed-stress iterations at the third refinement level (bottom right).
  • ...and 5 more figures

Theorems & Definitions (14)

  • Definition 1.1: The five-field formulation ahmed:hal-01687026
  • Definition 1.2: The space-time fixed-stress algorithm
  • Remark 3.3: The multi-rate FS
  • Remark 3.4: Space-time vs multi-rate
  • Remark 4.1: Algebraic errors
  • Remark 4.3: Space adaptivity
  • Remark 4.4: Static condensation
  • Remark 4.5: Computational cost
  • Lemma 5.1: Properties of $(\hat{p}^{k}_{h\tau},\hat{{\mathbf{u}}}^{k}_{h\tau})$
  • Theorem 6.1: Global-in-time a posteriori error estimate
  • ...and 4 more