Semi-explicit discretization schemes for weakly-coupled elliptic-parabolic problems

TL;DR

This paper proves first-order convergence of a semi-explicit Euler scheme combined with finite element discretization for weakly-coupled elliptic-parabolic problems, improving computational efficiency while maintaining accuracy.

Contribution

It introduces a decoupled semi-explicit scheme for elliptic-parabolic problems, providing convergence proof and explicit weak coupling condition quantification.

Findings

First-order convergence established for the scheme

Decoupling improves computational efficiency

Explicit weak coupling condition derived

Abstract

We prove first-order convergence of the semi-explicit Euler scheme combined with a finite element discretization in space for elliptic-parabolic problems which are weakly coupled. This setting includes poroelasticity, thermoelasticity, as well as multiple-network models used in medical applications. The semi-explicit approach decouples the system such that each time step requires the solution of two small and well-structured linear systems rather than the solution of one large system. The decoupling improves the computational efficiency without decreasing the convergence rates. The presented convergence proof is based on an interpretation of the scheme as an implicit method applied to a constrained partial differential equation with delay term. Here, the delay time equals the used step size. This connection also allows a deeper understanding of the weak coupling condition, which we accomplish to quantify explicitly.