On the stability of the ale space-time discontinuous Galerkin method for nonlinear convection-diffusion problems in time-dependent domains

Abstract

The paper is concerned with the analysis of the space-time discontinuous Galerkin method (STDGM) applied to the numerical solution of the nonstationary nonlinear convection-diffusion initial-boundary value problem in a time-dependent domain formulated with the aid of the arbitrary Lagrangian-Eulerian (ALE) method. In the formulation of the numerical scheme we use the nonsymmetric, symmetric and incomplete versions of the space discretization of diffusion terms and interior and boundary penalty. The nonlinear convection terms are discretized with the aid of a numerical flux. The space discretization uses piecewise polynomial approximations of degree not greater than p with an integer p ⩾ 1. In the theoretical analysis, the piecewise linear time discretization is used. The main attention is paid to the investigation of unconditional stability of the method.