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Error Analysis of the Explicit Splitting Scheme for Fluid-Poroelastic Structure Interaction Problems

Yifan Wang, Jeonghun Lee, Suncica Canic

Abstract

We present a priori error analysis for a fully discrete, parallelizable, explicit loosely coupled scheme for the time-dependent Stokes-Biot problem. The method decouples the fluid and poroelastic subproblems in a fully explicit fashion, allowing each problem to be solved independently at each time step, with a consistent treatment of the interface conditions that provides stability and convergence of the scheme. The error analysis is carried out in a discrete energy framework. More specifically, we introduce Ritz-type projections in each subdomain, and subtract the fully discrete scheme from the time-discrete continuous formulation. This yields reduced error equations in which the dominant interpolation contributions cancel. The remaining consistency terms stem primarily from time discretization residuals and lagged interface data inherent to the explicit splitting. The main result of this manuscript is the derivation of a discrete error energy identity, and establishment of unconditional error estimates in a combined energy-dissipation norm via a Gronwall-type argument. These estimates demonstrate first-order accuracy in time and optimal spatial convergence rates, as determined by the degree of the finite element polynomials. Numerical experiments based on a manufactured solution corroborate the theory, confirming first-order temporal convergence for all variables, and spatial convergence orders consistent with the chosen approximation spaces.

Error Analysis of the Explicit Splitting Scheme for Fluid-Poroelastic Structure Interaction Problems

Abstract

We present a priori error analysis for a fully discrete, parallelizable, explicit loosely coupled scheme for the time-dependent Stokes-Biot problem. The method decouples the fluid and poroelastic subproblems in a fully explicit fashion, allowing each problem to be solved independently at each time step, with a consistent treatment of the interface conditions that provides stability and convergence of the scheme. The error analysis is carried out in a discrete energy framework. More specifically, we introduce Ritz-type projections in each subdomain, and subtract the fully discrete scheme from the time-discrete continuous formulation. This yields reduced error equations in which the dominant interpolation contributions cancel. The remaining consistency terms stem primarily from time discretization residuals and lagged interface data inherent to the explicit splitting. The main result of this manuscript is the derivation of a discrete error energy identity, and establishment of unconditional error estimates in a combined energy-dissipation norm via a Gronwall-type argument. These estimates demonstrate first-order accuracy in time and optimal spatial convergence rates, as determined by the degree of the finite element polynomials. Numerical experiments based on a manufactured solution corroborate the theory, confirming first-order temporal convergence for all variables, and spatial convergence orders consistent with the chosen approximation spaces.
Paper Structure (22 sections, 6 theorems, 94 equations, 2 tables)

This paper contains 22 sections, 6 theorems, 94 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Continuous Problem
  3. Governing equations
  4. Interface conditions
  5. Weak formulation
  6. The Decoupled Continuous Stokes–Biot System
  7. Discretized problem: A parallelizable explicit splitting scheme
  8. Ritz Projections and General Approximation Properties
  9. Ritz projection for Stokes velocity and pressure:
  10. Ritz projection for Biot structure displacement
  11. Ritz projection for Biot pore pressure
  12. Trace and approximation inequalities
  13. Error Equations
  14. Error decomposition
  15. Subtraction of continuous and discrete problems
  16. ...and 7 more sections

Key Result

Lemma 4.1

\newlabelBoundaryTerms0 Let $\Omega_f \subset \mathbb{R}^d$ be a Lipschitz domain and let $\Gamma \subset \partial \Omega_f$ denote the fluid structure interface with unit outward normal $\boldsymbol{n}_f$. Then, for all $\boldsymbol{v} \in [H^1(\Omega_f)]^d$, the tangential projector $\boldsymbol Consequently, for all $\boldsymbol{w},\boldsymbol{v} \in [H^1(\Omega_f)]^d$, there exists a constant

Theorems & Definitions (9)

  • Lemma 4.1: Trace boundedness and continuity of the interface penalty terms
  • Lemma 4.2: Well-posedness and boundedness of the fluid Ritz bilinear form
  • Theorem 7.1: Estimates of residual terms
  • Proof 2
  • Lemma 7.2: Estimate of $Z_{n}$
  • Proof 3
  • Lemma 7.3
  • Theorem 7.4: A priori error estimate
  • Proof 4