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.

中文翻译：

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具有动态边界条件的抛物线问题的体表面李分裂

摘要 本文研究了具有动态边界条件的（半线性）抛物型偏微分方程的一阶体面分裂方法。所提出的李分裂方案是基于将问题重新表述为耦合的偏微分代数方程组，即边界条件被认为是耦合到体问题的第二个动态方程。分裂方法与体表面有限元和两个子系统的隐式欧拉离散化相结合。我们证明了在存在形式为 $\tau \leqslant ch$ 的弱 CFL 条件下，对于某个常数 $c>0$，得到的完全离散方案的一阶收敛性。收敛也使用艾伦卡恩类型的动态边界条件进行数值说明。