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A new operator splitting method for coupled PDE problems, with an application to poromechanics

Francesca Marcon, Stefano Scialò

arXiv:2609.31320v1math.NAcs.NA

TL;DR

Large discrete poromechanics systems motivate methods that decouple flow from mechanics. This paper reformulates the coupled problem as an optimization-based operator split, then reports faster iteration convergence in long-time simulations than the fixed-stress split. Its presentation uses quasi-static two-field Biot in 2D, while stating that the methodology is formulation-independent.

  • Problem

    The large discrete system in poromechanics motivates decoupling the flow and mechanics subproblems.

    Source: §1 · p. 1

  • Method

    The method introduces copies of shared unknowns and minimizes their mismatch with the originals under the system equations, enabling independent subproblem solves.

    Source: §1 · p. 1, §2 · p. 4

  • Results

    In long-time simulations, the proposed method converges in fewer iterations; on the tested meshes it needs more iterations than fixed-stress only during the first few time steps, then converges in a few iterations per step.

    Source: §3 · p. 6, §4 · p. 8

  • Takeaways & Limitations

    The method permits different meshes for different unknowns, offering flexible approximation choices, and is suitable for parallel computing.

    Source: §1 · p. 1, §4 · p. 8

    The presentation considers quasi-static two-field Biot in a two-dimensional domain for simplicity, although the authors state the methodology is independent of the chosen formulation.

    Source: §1 · p. 1

Abstract

from arXiv · show

The present work proposes a new operator splitting technique. The method leverages ideas used in PDE-constrained domain decomposition, adapting them to the objective of decoupling equations defined on the same geometric domain. The methodology is applied here to the resolution of coupled flow and deformation in the linear quasi-static Biot problem. An overview of its main characteristics is provided, also in comparison with the well established fixed-stress split for poroelasticity.

1. Introduction

The paper introduces an operator-splitting method based on PDE-constrained optimization, using additional variables and a mismatch functional to decouple coupled equations. It presents the method for poroelastic flow and deformation, alongside the fixed-stress split as a reference.

1. Introduction

  • The method copies unknowns shared across equations into additional variables and minimizes their mismatch with the originals, subject to the system equations.A gradient-based approach solves the system equations independently.Source: §1 · p. 1
  • The resulting smaller subproblems can be solved simultaneously, and independent discretizations allow different approximation levels for the variables.The paper identifies parallel computing and simulation efficiency and effectiveness as benefits of this flexibility.Source: §1 · p. 1
  • The approach is presented for coupled flow and deformation in poroelastic materials, while the proposed method is described as applicable to general coupled problems.The paper considers the quasi-static two-field Biot problem in a two-dimensional domain for simplicity, while stating that the methodology is independent of the chosen formulation.Source: §1 · p. 1
  • The fixed-stress split is the established reference: it solves flow using the previous iteration’s displacement, then updates displacement.Its convergence is studied in.Source: §1 · p. 1

2. Parallel operator splitting

The method augments the time-step problem with auxiliary unknowns and formulates its discrete solution as a minimization problem. It permits independent discretizations and parallel computation of the primal and dual problems.

2. Parallel operator splitting

  • At each time step, auxiliary unknowns Ψ∇·u and Ψp are introduced alongside displacement and pressure.Their time-step dependence is omitted from the notation.Source: §2 · p. 2
  • Independent discretizations may be used for the variables, with separate meshes for displacement, flow, and the auxiliary fields.The passage assigns mesh parameters hm, hf, h∇·u, and hp to these discretizations.Source: §2 · p. 2
  • Linear Lagrangian finite elements approximate displacement and pressure, while piecewise constants approximate Ψ∇·u and Ψp.Other approximation choices are also possible.Source: §2 · p. 3
  • The discrete algebraic problem collects basis-function integrals into matrices and represents unknowns as degree-of-freedom arrays, with b and g collecting known terms.The coefficient η scales the functional terms to help balance them during minimization.Source: §2 · p. 3
  • Linearity of the constraints allows the constrained minimization problem to be recast as an unconstrained minimization problem.The resulting problem is solved by a gradient-based approach.Source: §2 · pp. 3–4
  • The gradient direction at the initial auxiliary-variable values is computed using Algorithm 1.The initial search-direction procedure solves primal and dual problems after computing initial displacement and pressure.Source: §2 · p. 4
  • The descent direction can be computed by solving the primal and dual problems independently in parallel; for self-adjoint operators, those problems coincide.The paper notes that efficient solvers for the decoupled problems can make the splitting more convenient.Source: §2 · p. 4

Fixed stress split

The fixed-stress scheme is reported as the reference formulation for comparison in the numerical results. Its description indexes the procedure by time step and iteration and identifies E as the drained bulk modulus.

Fixed stress split

  • The fixed-stress formulation is reported as the reference scheme for comparison in the numerical results.The section refers to the formulation used for that comparison.Source: Fixed stress split · p. 4
  • The scheme uses time-step index k and iteration index i, with prior displacement and pressure iterates given.The passage specifies uk−1 and (ph)k,i−1 as given.Source: Fixed stress split · p. 4
  • E denotes the drained bulk modulus in the fixed-stress formulation.Source: Fixed stress split · p. 4

3. Numerical Results

In the Terzaghi test, POS pressure errors are close to the fixed-stress and monolithic results, and POS needs fewer iterations after its initial steps. Varying mesh choices offers accuracy and computational-cost trade-offs, while auxiliary-mesh changes have limited effects on error and iteration count.

3. Numerical Results

  • The Terzaghi test uses a homogeneous 1 m square, top-edge compression, lateral vertical-only displacement, bottom clamping, and top Dirichlet/rest Neumann flow boundaries.Four potentially independent triangular meshes resolve the problem.Source: §3 · pp. 4–5
  • FS and monolithic pressure-error curves overlap, while POS lies slightly below; POS needs more iterations only initially, then converges in a few per step.FS iteration counts remain fairly constant as time advances.Source: Figure 1 · p. 5, §3 · p. 6
  • POS takes about twice FS’s average time per iteration and has higher preprocessing costs, but its lower iteration count can make it convenient for long integrations.The POS primal and dual linear systems and its preprocessing can be solved in parallel.Source: Table 1 · p. 5, §3 · p. 6
  • Changing the mechanics and flow meshes offers intermediate accuracy and computational-cost levels; matching all meshes reduces preprocessing time, while iteration counts vary little.The preprocessing increase for differing meshes remains limited.Source: §3 · p. 8
  • With mechanics and flow meshes fixed at 0.01, auxiliary-mesh changes most affect preprocessing; error and iteration count change only slightly.Time per iteration rises marginally with auxiliary-mesh refinement, and these mesh choices relate to data shared in parallel implementations.Source: §3 · p. 8

4. Conclusions

The conclusion presents an optimization-based operator split for linear quasi-static poroelasticity and compares it with fixed stress. In this preliminary investigation, POS converges in fewer iterations for long simulations and supports flexible meshes and parallel implementation.

4. Conclusions

  • The method reformulates the original problem as an optimization problem and is presented as suitable for a large variety of applications.It is applied to the linear, quasi-static Biot problem and compared with the fixed-stress split.Source: §4 · p. 8
  • For long-time simulations, the method shows faster convergence in iteration count.Source: §4 · p. 8
  • Different meshes can be used for different unknowns, allowing flexible approximation choices.Source: §4 · p. 8
  • The method is suitable for parallel implementation.Source: §4 · p. 8
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