Well-posedness of hp -version discontinuous Galerkin methods for fractional diffusion wave equations

We establish the well-posedness of an hp-version time-stepping discontinuous Galerkin method for the numerical solution of fractional superdiffusion evolution problems. In particular, we prove the existence and uniqueness of approximate solutions for generic hp-version finite element spaces featuring nonuniform time steps and variable approximation degrees. We then derive new hp-version error estimates in a nonstandard norm, which are completely explicit in the local discretization and regularity parameters. As a consequence, we show that by employing geometrically refined time steps and linearly increasing approximation orders, exponential rates of convergence in the number of temporal degrees of freedom are achieved for solutions with singular (temporal) behaviour near t=0 caused by the weakly singular kernel. Moreover, we show optimal algebraic convergence rates for h-version approximations on graded meshes. We present a series of numerical tests where we verify experimentally that our theoretical convergence properties also hold true in the stronger L_∞ norm.