A posteriori analysis for space-time, discontinuous in time Galerkin approximations for parabolic equations in a variable domain

¹ Department of Mathematics and Applied Mathematics, University of Crete, 714 09 Heraklion, Greece
² Department of Mathematics, University of Chester, Thornton Science Park, CH2 4NU, UK
³ Institute of Applied and Computational Mathematics, FORTH, 711 10 Heraklion, Greece

^* Corresponding author: d.antonopoulou@chester.ac.uk

Received: 28 December 2016
Accepted: 25 September 2018

Abstract

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 Jamet (SIAM J. Numer. Anal. 15 (1978) 913–928). Using a Clément-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 coefficients 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 Evans (American Mathematical Society (1998)) 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 (Comput. Meth. Appl. Mech. Eng. 167 (1998) 223–237), proposed for adaptive, Runge-Kutta finite element methods for linear parabolic problems. Our theoretical results are verified by numerical experiments.