Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Convergence analysis and a novel Lagrange multiplier partitioned method for fluid-poroelastic interaction

Amy de Castro, Hyesuk Lee

TL;DR

This work develops a strongly coupled, non-iterative partitioned method for the fully dynamic Stokes-Biot FPSI problem by employing three Lagrange multipliers to impose interface conditions and formulating a Schur-complement equation for the interfacial unknowns. The authors prove spatial convergence of the finite-element discretization and present an efficient, matrix-free strategy for applying the Schur operator, together with a purpose-built preconditioner that enables fast, parallelizable subdomain solves. Numerical tests with a manufactured solution confirm the expected convergence rates, while a hydrological application demonstrates robustness under challenging parameters and highlights the essential role of the preconditioner in guaranteeing convergence. Collectively, the approach enables accurate, scalable simulation of fluid-poroelastic interactions by decoupling subproblems after a Schur solve, with practical impact for geophysical and biomedical applications where FPSI phenomena occur.

Abstract

We propose a partitioned method for the monolithic formulation of the Stokes-Biot system that incorporates Lagrange multipliers enforcing the interface conditions. The monolithic system is discretized using finite elements, and we establish convergence of the resulting approximation. A Schur complement based algorithm is developed together with an efficient preconditioner, enabling the fluid and poroelastic structure subproblems to be decoupled and solved independently at each time step. The Lagrange multipliers approximate the interface fluxes and act as Neumann boundary conditions for the subproblems, yielding parallel solution of the Stokes and Biot equations. Numerical experiments demonstrate the effectiveness of the proposed algorithm and validate the theoretical error estimate.

Convergence analysis and a novel Lagrange multiplier partitioned method for fluid-poroelastic interaction

TL;DR

This work develops a strongly coupled, non-iterative partitioned method for the fully dynamic Stokes-Biot FPSI problem by employing three Lagrange multipliers to impose interface conditions and formulating a Schur-complement equation for the interfacial unknowns. The authors prove spatial convergence of the finite-element discretization and present an efficient, matrix-free strategy for applying the Schur operator, together with a purpose-built preconditioner that enables fast, parallelizable subdomain solves. Numerical tests with a manufactured solution confirm the expected convergence rates, while a hydrological application demonstrates robustness under challenging parameters and highlights the essential role of the preconditioner in guaranteeing convergence. Collectively, the approach enables accurate, scalable simulation of fluid-poroelastic interactions by decoupling subproblems after a Schur solve, with practical impact for geophysical and biomedical applications where FPSI phenomena occur.

Abstract

We propose a partitioned method for the monolithic formulation of the Stokes-Biot system that incorporates Lagrange multipliers enforcing the interface conditions. The monolithic system is discretized using finite elements, and we establish convergence of the resulting approximation. A Schur complement based algorithm is developed together with an efficient preconditioner, enabling the fluid and poroelastic structure subproblems to be decoupled and solved independently at each time step. The Lagrange multipliers approximate the interface fluxes and act as Neumann boundary conditions for the subproblems, yielding parallel solution of the Stokes and Biot equations. Numerical experiments demonstrate the effectiveness of the proposed algorithm and validate the theoretical error estimate.
Paper Structure (10 sections, 3 theorems, 63 equations, 5 figures, 3 tables)

This paper contains 10 sections, 3 theorems, 63 equations, 5 figures, 3 tables.

Table of Contents

  1. Background and Research Goals
  2. Spatial convergence analysis
  3. Partitioned Method
  4. Efficient Computation of the Schur Complement Equation
  5. Matrix-Vector Products with $S$
  6. Preconditioner for $S$
  7. Numerical Results
  8. FEM-FEM coupling
  9. Hydrological Example
  10. Conclusions

Key Result

Theorem 2.1

Let $(\bm{u}^{n+1}, p_{f}^{n+1}, \bm{\eta}^{n+1}, p_{p}^{n+1}, g_{1}^{n+1}, g_{2}^{n+1}, \lambda_{p}^{n+1}) \in A$ be the solution to the semi-discrete system FPSI_FEM:WF:allTimeDisc_StabError with $\bm{u}_N^{n+1}=\bm{\eta}_N^{n+1} =\bm{0}$. Assume that this solution has sufficient regularity. Then

Figures (5)

  • Figure 1: Manufactured solution: relative residual versus iteration number for BiCGstab(l).
  • Figure 2: Fluid-poroelastic domain for hydrological example
  • Figure 3: Hydrological example, case 1: all physical parameters set to 1.
  • Figure 4: Hydrological example, case 2: $\kappa = 10^{-4}, s_0 = 10^{-4}, \lambda = 10^6$. Other parameters set to 1.
  • Figure 5: Hydrological example, case 2: relative residual versus iteration number for BiCGstab(l).

Theorems & Definitions (5)

  • Theorem 2.1
  • proof
  • Lemma 2.2: Discrete Gronwall's lemma
  • Corollary 2.3
  • Remark