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

TL;DR

This paper provides a rigorous error analysis for an explicit, decoupled numerical scheme solving fluid-poroelastic structure interaction problems, demonstrating stability, convergence, and optimal accuracy in both time and space.

Contribution

It introduces a novel a priori error analysis framework for a fully explicit, parallelizable scheme for the Stokes-Biot problem, establishing unconditional error estimates.

Findings

First-order accuracy in time for all variables

Optimal spatial convergence rates based on finite element degree

Numerical experiments confirm theoretical error estimates

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.