Bulk-surface Lie splitting for parabolic problems with dynamic boundary conditions

TL;DR

This paper introduces a first-order bulk-surface Lie splitting method for parabolic PDEs with dynamic boundary conditions, reformulating the problem as a coupled PDE-DAE system and proving convergence under a weak CFL condition.

Contribution

The paper proposes a novel Lie splitting scheme for parabolic problems with dynamic boundary conditions, combining finite elements and implicit Euler, with proven first-order convergence.

Findings

Proven first-order convergence of the scheme.

Numerical illustration with Allen-Cahn-type boundary conditions.

Scheme works under a weak CFL condition.

Abstract

This paper studies bulk-surface splitting methods of first order for (semi-linear) parabolic partial differential equations with dynamic boundary conditions. The proposed Lie splitting scheme is based on a reformulation of the problem as a coupled partial differential-algebraic equation system, i.e., the boundary conditions are considered as a second dynamic equation which is coupled to the bulk problem. The splitting approach is combined with bulk-surface finite elements and an implicit Euler discretization of the two subsystems. We prove first-order convergence of the resulting fully discrete scheme in the presence of a weak CFL condition of the form $\tau \leq c h$ for some constant $c>0$. The convergence is also illustrated numerically using dynamic boundary conditions of Allen-Cahn-type.