Discontinuous Galerkin Approximation to a Class of Strongly Damped Wave Equations with Uniform Exponential Decay
This paper proposes and examines a completely discrete discontinuous Galerkin finite element scheme to approximate the solution of a class of strongly damped wave equations with uniform exponential decay property. In the first part, a semidiscrete discontinuous Galerkin approximation is employed in the spatial variable, while keeping time variable continuous and optimal error estimates are established while preserving the exponential decay property of the original system. Subsequently, a fully discrete scheme is developed and analyzed. Using suitable energy techniques, the consistency between the decay properties of the continuous and discrete systems is demonstrated. In particular, the constants appearing in various estimates remain uniformly bounded even as the damping parameters approach zero. Moreover, error estimates that preserve the uniform decay property are highlights of this paper. Further extension of our analysis includes a class of strongly damped fourth order evolution equations using SIPG as well as $C^0$-IP methods. Finally, several numerical experiments are performed to confirm the theoretical convergence results, verify the predicted decay behavior, and investigate the influence of model parameters on stability and computational accuracy.