Volume 53, Number 2, March-April 2019
|Page(s)||523 - 549|
|Published online||24 April 2019|
A posteriori analysis for space-time, discontinuous in time Galerkin approximations for parabolic equations in a variable domain
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: email@example.com
Accepted: 25 September 2018
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.
Mathematics Subject Classification: 65M12 / 65M15 / 65M60
Key words: Heat equation / space-time discontinuous Galerkin methods / a posteriori error estimates / non-cylindrical domains
© EDP Sciences, SMAI 2019
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