A second-order bulk--surface splitting for parabolic problems with dynamic boundary conditions

TL;DR

This paper presents a second-order bulk--surface splitting scheme for semi-linear parabolic PDEs with dynamic boundary conditions, combining reformulation as a PDE-DAE, delay terms, finite elements, and BDF time discretization, with proven convergence and numerical validation.

Contribution

It introduces a novel second-order splitting scheme for parabolic PDEs with dynamic boundary conditions, including a reformulation as PDE-DAE and delay terms, with theoretical and numerical validation.

Findings

Proven second-order convergence under weak CFL condition.

Numerical experiments confirm theoretical convergence.

Potential for higher-order schemes demonstrated numerically.

Abstract

This paper introduces a novel approach for the construction of bulk--surface splitting schemes for semi-linear parabolic partial differential equations with dynamic boundary conditions. The proposed construction is based on a reformulation of the system as a partial differential--algebraic equation and the inclusion of certain delay terms for the decoupling. To obtain a fully discrete scheme, the splitting approach is combined with finite elements in space and a BDF discretization in time. Within this paper, we focus on the second-order case, resulting in a $3$-step scheme. We prove second-order convergence under the assumption of a weak CFL-type condition and confirm the theoretical findings by numerical experiments. Moreover, we illustrate the potential for higher-order splitting schemes numerically.