Discontinuous Galerkin Approximation of Linear Parabolic Problems with Dynamic Boundary Conditions

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 $$p\ge 1$$p?1 on meshes with granularity h along with a backward Euler time-stepping scheme with time-step $$\Delta t$$Δt, 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 $$h^p + \Delta t$$hp+Δt. The sharpness of the theoretical estimates are verified through several numerical experiments.