hp-version discontinuous Galerkin time-stepping schemes for diffusive-viscous wave equation

Abstract

This work introduces a fully discrete scheme for the diffusion-viscous wave equation (DVWe) in the second-order formulation, combining hp-DG time-stepping schemes with conforming finite element methods (FEM). Two major theoretical contributions are presented: (1) hp-version a priori error estimates in both the energy-norm and DG-norm, which are optimal in the spatial mesh size h, temporal step size τ, and temporal polynomial order q, yet suboptimal by one order in the spatial polynomial order p. Furthermore, for solutions exhibiting weak singularities in time, exponential convergence in terms of the total number of temporal degrees of freedom is proven using the hp-refinement strategy. (2) An energy decay estimate that offers explicit bounds involving the model parameters, discretization parameters, and the Poincaré inequality constant. A series of numerical experiments are presented to validate the practical performance of the proposed approach.