Discontinuous Galerkin approximation of linear parabolic problems with dynamic boundary conditions

TL;DR

This paper develops and analyzes a Discontinuous Galerkin method for linear parabolic problems with dynamic boundary conditions, providing stability, error estimates, and numerical validation of the approach.

Contribution

It introduces a novel DG scheme for such problems, proving stability and optimal error estimates with numerical verification.

Findings

The fully discrete scheme is stable and converges with optimal order.

Error estimates are sharp and verified numerically.

The method effectively handles dynamic boundary conditions in parabolic problems.

Abstract

In this paper we propose and analyze a Discontinuous Galerkin method for a linear parabolic problem with dynamic boundary conditions. We present the formulation and prove stability and optimal a priori error estimates for the fully discrete scheme. More precisely, using polynomials of degree $p\geq 1$ on meshes with granularity $h$ along with a backward Euler time-stepping scheme with time-step $\Delta t$, we prove that the fully-discrete solution is bounded by the data and it converges, in a suitable (mesh-dependent) energy norm, to the exact solution with optimal order $h^p + \Delta t$. The sharpness of the theoretical estimates are verified through several numerical experiments.