• A Posteriori Analysis for Space-Time, discontinuous in time Galerkin approximations for parabolic equations in a variable domain

      Antonopoulou, Dimitra; Plexousakis, Michael; University of Chester; University of Crete (ECP sciences, 2019-04-24)
      This paper presents an a posteriori error analysis for the discontinuous in time space-time scheme proposed by Jamet for the heat equation in multi-dimensional, non-cylindrical domains [25]. Using a Cl ement-type interpolant, we prove abstract a posteriori error bounds for the numerical error. Furthermore, in the case of two-dimensional spatial domains we transform the problem into an equivalent one, of parabolic type, with space-time dependent coe cients but posed on a cylindrical domain. We formulate a discontinuous in time space{time scheme and prove a posteriori error bounds of optimal order. The a priori estimates of [19] for general parabolic initial and boundary value problems are used in the derivation of the upper bound. Our lower bound coincides with that of Picasso [36], proposed for adaptive, Runge-Kutta finite element methods for linear parabolic problems. Our theoretical results are verified by numerical experiments.
    • A posteriori error estimates for fully discrete fractional-step ϑ-approximations for parabolic equations

      Karakatsani, Fotini; University of Chester (Oxford University Press, 2015-07-22)
      We derive optimal order a posteriori error estimates for fully discrete approximations of initial and boundary value problems for linear parabolic equations. For the discretisation in time we apply the fractional-step #-scheme and for the discretisation in space the finite element method with finite element spaces that are allowed to change with time.