A Priori $hp$-estimates for Discontinuous Galerkin Approximations to   Linear Hyperbolic Integro-Differential Equations
  An $hp$-discontinuous Galerkin (DG) method is applied to a class of second order linear hyperbolic integro-differential equations. Based on the analysis of an expanded mixed type Ritz-Volterra projection, {\it a priori} $hp$-error estimates in $L^{\infty}(L^2)$-norm of the velocity as well as of the displacement, which are optimal in the discretizing parameter $h$ and suboptimal in the degree of polynomial $p$ are derived. For optimal estimates of the displacement in $L^{\infty}(L^2)$-norm with reduced regularity on the exact solution, a variant of Baker's nonstandard energy formulation is developed and analyzed. Results on order of convergence which are similar in spirit to linear elliptic and parabolic problems are established for the semidiscrete case after suitably modifying the numerical fluxes. For the completely discrete scheme, an implicit-in-time procedure is formulated, stability results are derived and {\it a priori} error estimates are discussed. Finally, numerical experiments on two dimensional domains are conducted which confirm the theoretical results. 
Samir Karaa, Amiya K. Pani and Sangita Yadav
math.NA