Reading group

Firedrake reading group

An informal reading group around many things finite element. Hosted online. Email Lawrence Mitchell if you'd like to participate. Suggestions of papers and scheduling discussions also take place in the #readinggroup channel on the Firedrake slack.

Scheduled time: Mondays 17:00UTC. This calendar contains details.

Hosted via Zoom:

Meeting ID: 928 2384 2685
Passcode: 494089

Upcoming sessions

2021-05-10 Lawrence Mitchell leading Fast parallel solution of fully implicit Runge-Kutta and discontinuous Galerkin in time for numerical PDEs, Part II: nonlinearities and DAEs, Ben S. Southworth, Oliver Krzysik, Will Pazner https://arxiv.org/pdf/2101.01776.pdf
2021-05-03 No session, UK bank holiday
In April, if there's interest, Alberto Paganini leading Finite Elements and Virtual Elements on Classical Meshes, Franco Brezzi and Luisa Donatella Marini, https://link.springer.com/article/10.1007%2Fs10013-021-00474-y

Possible papers

Please add your own, or claim one to present.

Bonizzoni and Kanschat, H¹-conforming finite element cochain complexes and commuting quasi-interpolation operators on cartesian meshes (2020).
Li, Zhang, and Zhang, A constrained transport divergence-free finite element method for Incompressible MHD equations (2020).
Xu, The method of subspace corrections (2001).
Hiptmair and Pechstein, Discrete regular decompositions of tetrahedral discrete 1-forms (2017). This paper introduces, in a rather didactic way, a lot of modern analysis tools for development and analysis of preconditioners for H(div) and H(curl) problems.

Past editions

2021-04-26 Lawrence Mitchell leading Fast parallel solution of fully implicit Runge-Kutta and discontinuous Galerkin in time for numerical PDEs, Part I: the linear setting, Ben S. Southworth, Oliver Krzysik, Will Pazner, Hans De Sterck https://arxiv.org/pdf/2101.00512.pdf might be reasonably be implementable as a PC inside irksome.

2021-04-12 & 2021-04-19 Alberto Paganini went through An overview on deep learning-based approximation methods for partial differential equations, Christian Beck, Martin Hutzenthaler, Arnulf Jentzen, and Benno Kuckuck https://arxiv.org/pdf/2012.12348.pdf This mathematical paper summarises known theorems about approximation of PDEs using neural networks. It also has a substantial bibliography with pointers to applications in many areas.

2021-03-15 Francis Aznaran went through Positivity-preserving methods for population models, Sergio Blanes, Arieh Iserles, Shev Macnamara https://arxiv.org/pdf/2102.08242.pdf it turns out to be quite tricky, approximations of operator exponentials and Padé approximations were at the forefront.

2021-03-08 Thomas Gibson and Jack Betteridge went through Multigrid preconditioners for the hybridized Discontinuous Galerkin discretisation of the shallow water equations, Jack D. Betteridge, Thomas H. Gibson, Ivan G. Graham, Eike H. Mueller https://arxiv.org/pdf/2004.09389.pdf

2021-02-22 Alexei Gazca went over Multilevel methods for nonuniformly elliptic operators, Long Chen, Ricardo H. Nochetto, Enrique Otarola, Abner J. Salgado https://arxiv.org/pdf/1403.4278. You can do fractional laplacians with one extra space dimension (should be able to do 3D+1 in Firedrake with a few additions in UFL for geometry terms).

2021-02-15 Matt Knepley went over Accurate Discretization Of Poroelasticity Without Darcy Stability -- Stokes-Biot Stability Revisited, Kent-Andre Mardal, Marie E. Rognes, Travis B. Thompson https://arxiv.org/pdf/2007.10012.pdf the headline message is that you can use more element pairs than you thought you could and still have a parameter-robust discretisation.

2021-02-01 Lawrence Mitchell went over Space-time block preconditioning for incompressible flow, Danieli, Southworth, Wathen https://arxiv.org/abs/2101.07003

2020-12-14 David Ham went over a proposal to properly add dual spaces in UFL, and the consequences thereof. Now available on the arxiv.

2020-12-07 Colin Cotter covered some of the (relatively new) developments in parallel in time stuff, going over the introduction to ParaDIAG from Martin Gander and friends.

2020-11-30 Florian Wechsung covered one paper on the new hotness of using neural nets for everything: Zongyi Li, Nikola Kovachki, Kamyar Azizzadenesheli, Burigede Liu, Kaushik Bhattacharya, Andrew Stuart, Anima Anandkumar, Fourier Neural Operator for Parametric Partial Differential Equations (2020)

2020-11-23 Florian Wechsung covered the (now sadly bitrotted) automated adaptivity that used to be implemented in Dolfin, described in Marie E. Rognes and Anders Logg, Automated Goal-Oriented Error Control I: Stationary Variational Problems (2013).

2020-11-09 Eder Medina went through Mardal and Winther, Preconditioning discretizations of systems of partial differential equations (2011).

2020-10-26 & 2020-11-02 Ioannis Papadopoulos talked us through the construction and analysis of the Stokes-Darcy element introduced in Mardal, Tai, and Winther, A robust finite element method for Darcy-Stokes flow (2002). (Available in Firedrake as FunctionSpace(triangular_mesh, "MTW", 3)).

2020-10-12 & 2020-10-19. Colin Cotter talked us through the main points of Demkowicz and Gopalakrishnan, An overview of the discontinuous Petrov Galerkin method (2014).

Summer 2020. We read through Joachim Schöberl's thesis on parameter-robust multigrid methods. Some notes are collected.

Autumn/Winter 2019. We read Doug Arnold's book on Finite Element Exterior Calculus.