Speaker: William McLean, University of New South Wales

Time : 10:00 - 11:00 CEST (Rome/Paris)

Hosted at: SISSA, International School of Advanced Studies, Trieste, Italy

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Organizers : Pavan Pranjivan Mehta* (pavan.mehta@sissa.it) and Arran Fernandez** (arran.fernandez@emu.edu.tr)

* SISSA, International School of Advanced Studies, Italy

** Eastern Mediterranean University, Northern Cyprus

Keywords: Time discretisation, Gauss--Radau quadrature, Superconvergence, Reconstruction, Radau IIA Runge--Kutta method, Error profile

Abstract: Discontinuous Galerkin (DG) time stepping is an alternative to popular schemes for fractional-order problems, such as convolution quadrature and the L1 method (and variants thereof). The talk will be an introduction to DG and its properties, including
superconvergence behaviour at the local Radau points. The numerical analysis for DG schemes applied to classical ODEs and to parabolic PDEs is well-developed [1, 2, 3, 5, 8], but plenty of open questions remain for subdiffusion equations [6, 7]

Biography: William McLean is an Associate Professor in the School of Mathematics and Statistics at the University of New South Wales (UNSW). He completed his PhD at the Australian National University in 1985, and held positions at Oregon State University and the University of Tasmania before moving to UNSW in 1989. With research interests in numerical analysis for integral equations and fractional PDEs, he has authored over 60 papers and one monograph [4], as well as serving on the editorial boards of the ANZIAM Journal, the Journal of Integral Equations and the SIAM Journal on Numerical Analysis.

Bibliography

[1] P. Lesaint and P. A. Raviart, On a finite element method for solving the neutron transport equation, Publications Math´ematiques et Informatiques de Rennes, Journ´ees´el´ements finis, no. S4, 1974.
[2] M. Delfour, W. Hager and F. Trochu, Discontinous Galerkin methods for ordinary differential equations, Math. Comp., 36: 455–473, 1981.
[3] Kenneth Eriksson, Claes Johnson and Vidar Thom´ee, Time discretisation of parabolic problems by the discontinuous Galerkin method, M2AN, 19: 611–642, 1985.
[4] William McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, 2000.
[5] Vidar Thomee, Galerkin Finite Element Methods for Parabolic Problems (Second Edition), Springer 2006.

[6] Kassem Mustapha and William McLean, Superconvergence of a discontinuous Galerkin method for fractional diffusion and wave equations, SIAM J. Numer. Anal., 51: 491–515.
[7] William McLean, Implementation of high-order discontinuous Galerkin time stepping for fractional diffusion problems, ANZIAM J., 62: 121–147, 2020.
[8] Norikazu Saito, Variational analysis of the discontinuous Galerkin time-stepping method for parabolic equations, IMA J. Numer. Anal., 41: 1267–1292, 2021.