Numerical analysis of parabolic problems with dynamic boundary conditions

TL;DR

This paper analyzes numerical methods for parabolic PDEs with dynamic boundary conditions, focusing on discretization, error analysis, and efficient time integration techniques for complex boundary interactions.

Contribution

It introduces a unified weak formulation for parabolic problems with dynamic boundaries, including error analysis and mass lumping techniques for improved numerical solutions.

Findings

Error estimates for finite element discretizations on polygonal and smooth domains.

Effective use of mass lumping with exponential integrators.

Validation of the framework for coupled bulk-surface reaction-diffusion problems.

Abstract

Space and time discretizations of parabolic differential equations with dynamic boundary conditions are studied in a weak formulation that fits into the standard abstract formulation of parabolic problems, just that the usual L^2(\Omega) inner product is replaced by an L^2(\Omega) \oplus L^2(\Gamma) inner product. The class of parabolic equations considered includes linear problems with time- and space-dependent coefficients and semilinear problems such as reaction-diffusion on a surface coupled to diffusion in the bulk. The spatial discretization by finite elements is studied in the proposed framework, with particular attention to the error analysis of the Ritz map for the elliptic bilinear form in relation to the inner product, both of which contain boundary integrals. The error analysis is done for both polygonal and smooth domains. We further consider mass lumping, which enables us to use exponential integrators and bulk-surface splitting for time integration.