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Higher-order iterative decoupling for poroelasticity

Robert Altmann, Abdullah Mujahid, Benjamin Unger

TL;DR

This work addresses the computational challenge of solving coupled poroelasticity equations, formulated as an elliptic–parabolic PDE system, via higher-order time discretization and iterative decoupling. It combines fixed-stress operator splitting with BDF-$k$ time integration ($k\le 5$) and derives a contractivity condition with stabilization $L$ that ensures convergence, yielding a rigorous error bound that balances temporal discretization and iteration tolerance. A key practical contribution is the explicit guideline ${\rm TOL} \approx C\tau^{k+3/2}$ to minimize total error, validated through convergence studies, error balancing experiments, and a real-world biomechanics brain network model. The results offer actionable insights for efficiently solving high-order decoupled poroelastic problems in 2D/3D with potential extensions to Runge–Kutta time-stepping for broader applicability.

Abstract

For the iterative decoupling of elliptic-parabolic problems such as poroelasticity, we introduce time discretization schemes up to order $5$ based on the backward differentiation formulae. Its analysis combines techniques known from fixed-point iterations with the convergence analysis of the temporal discretization. As main result, we show that the convergence depends on the interplay between the time step size and the parameters for the contraction of the iterative scheme. Moreover, this connection is quantified explicitly, which allows for balancing the single error components. Several numerical experiments illustrate and validate the theoretical results, including a three-dimensional example from biomechanics.

Higher-order iterative decoupling for poroelasticity

TL;DR

This work addresses the computational challenge of solving coupled poroelasticity equations, formulated as an elliptic–parabolic PDE system, via higher-order time discretization and iterative decoupling. It combines fixed-stress operator splitting with BDF- time integration () and derives a contractivity condition with stabilization that ensures convergence, yielding a rigorous error bound that balances temporal discretization and iteration tolerance. A key practical contribution is the explicit guideline to minimize total error, validated through convergence studies, error balancing experiments, and a real-world biomechanics brain network model. The results offer actionable insights for efficiently solving high-order decoupled poroelastic problems in 2D/3D with potential extensions to Runge–Kutta time-stepping for broader applicability.

Abstract

For the iterative decoupling of elliptic-parabolic problems such as poroelasticity, we introduce time discretization schemes up to order based on the backward differentiation formulae. Its analysis combines techniques known from fixed-point iterations with the convergence analysis of the temporal discretization. As main result, we show that the convergence depends on the interplay between the time step size and the parameters for the contraction of the iterative scheme. Moreover, this connection is quantified explicitly, which allows for balancing the single error components. Several numerical experiments illustrate and validate the theoretical results, including a three-dimensional example from biomechanics.
Paper Structure (12 sections, 4 theorems, 70 equations, 4 figures, 5 tables)

This paper contains 12 sections, 4 theorems, 70 equations, 4 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Preliminaries and Problem Formulation
  3. Abstract formulation as elliptic--parabolic problem
  4. BDF operators and weighted inner product spaces
  5. Discrete Grönwall-type inequality
  6. Iterative Decoupling and BDF Time Integration
  7. Numerical Experiments
  8. Convergence studies
  9. Error balancing
  10. Needed iteration steps to reach TOL
  11. Biomechanics multiple-network problem
  12. Conclusions

Key Result

Proposition 2.2

Let $\mathcal{X}$ be a Hilbert space and $k\in\{1,\ldots,5\}$. Then there exist a parameter $\eta\in[0,1)$, a matrix $G = [g_{ij}]\in\mathbb{S}_{\succ}^{k}$, and a vector $\gamma\in\mathbb{R}^{k+1}$ such that for any self-adjoint and monotone operator $\mathcal{M}\colon\mathcal{X}\to\mathcal{X}$, an

Figures (4)

  • Figure 1: Errors for BDF-$k$ for $1 \le k \le 5$ for a fixed spatial mesh with mesh size $h=2^{-6}$. The solid blue curves correspond to the newly introduced decoupled methods whereas the dashed red curves correspond to the fully coupled methods (original BDF schemes). Left: $\mathcal{V}$-error in the displacement $u$. Right: $\mathcal{H}_{\hbox{$\mathcal{Q}$}}$-error in the pressure $p$.
  • Figure 2: Errors for BDF-$1$ (left), BDF-$2$ (middle), and BDF-$3$ (right) for a fixed spatial mesh size $h=2^{-7}$. The gray line indicate orders $1$ (left), $2$ (middle), and $3$ (right).
  • Figure 3: (a) The red part corresponds to ventricular boundary $\partial\Omega_v$, whereas the outer boundary is $\partial\Omega_s$. (b) Convergence history for all four fields.
  • Figure 4: Illustration of the deformation and the three pressure variables at the final time $T=1$.

Theorems & Definitions (13)

  • Example 2.1: multiple network case
  • Proposition 2.2
  • Remark 2.3
  • proof : Proof of \ref{['prop:summation']}
  • Example 2.4
  • Lemma 2.5
  • proof
  • Lemma 3.1
  • proof
  • Theorem 3.2: convergence result
  • ...and 3 more