Decoupling multistep schemes for elliptic-parabolic problems

TL;DR

This paper develops higher-order decoupling multistep schemes for elliptic-parabolic problems, including multi-network poroelasticity, with a new convergence proof based on G-stability and weighted norms.

Contribution

It introduces a modified scheme with a novel convergence proof applicable to any order, improving understanding of weak coupling conditions.

Findings

Convergence proof valid for any order of the scheme.

Sharper characterization of weak coupling conditions.

Effective use of weighted norms for error analysis.

Abstract

We study the construction and convergence of decoupling multistep schemes of higher order using the backward differentiation formulae for an elliptic-parabolic problem, which includes multiple-network poroelasticity as a special case. These schemes were first introduced in [Altmann, Maier, Unger, BIT Numer. Math., 64:20, 2024], where a convergence proof for the second-order case is presented. Here, we present a slightly modified version of these schemes using a different construction of related time delay systems. We present a novel convergence proof relying on concepts from G-stability applicable for any order and providing a sharper characterization of the required weak coupling condition. The key tool for the convergence analysis is the construction of a weighted norm enabling a telescoping argument for the sum of the errors.