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 (DG) method for the numerical solution of fractional super-diffusion evolution problems. In particular, we prove the existence and uniqueness of approximate solutions for generic hp-version finite element spaces featuring non-uniform time-steps and variable approximation degrees. We then derive new hp-version error estimates in a non-standard 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) behavior 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.

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