Bulk–surface Lie splitting for parabolic problems with dynamic boundary conditions. (18th March 2022)

• Record Type: Journal Article
• Authors: Altmann, Robert; Kovács, Balázs; Zimmer, Christoph
• Is Part Of: IMA journal of numerical analysis. Volume 43:Number 2(2023)
• Page Start: 950
• Page End: 975
• Publication Date: 2022-03-18
• Subjects: dynamic boundary conditions -- PDAE -- splitting methods -- bulk–surface splitting -- parabolic equations; Numerical analysis -- Periodicals; 519.405
• DOI: 10.1093/imanum/drac002
• Languages: English
• ISSNs: 0272-4979

Abstract

Abstract: This paper studies bulk–surface splitting methods of first order for (semilinear) 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 that 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 $\tau \leqslant c h$ for some constant $c>0$ . The convergence is also illustrated numerically using dynamic boundary conditions of Allen–Cahn type.