Irish Numerical Analysis Forum The 17th Workshop on Numerical Methods for Problems with Layer Phenomena was the first in this series to be held online instead of physically. Its success opened our eyes to the possibility of organising talks by speakers located in any part of the globe. Thus, in collaboration with other Irish researchers, we have now created the (virtual) Irish Numerical Analysis Forum which will include fortnightly seminars in all areas of numerical analysis that are aligned with the interests of the Irish numerical analysis community. Its aim will be to solicit lectures from leading international numerical analysts who will discuss their research area in a style that is accessible to most numerical analysts (i.e., not just those who are already familiar with the subject of the lecture). The seminar series will start in January 2021. The talks will be streamed online via zoom and are free to view; one must however register in advance with INAF to gain access to them. We will usually have one talk every two weeks, but the talk timetable may vary from this. A registration puts you on our mailing list for receiving zoom links for all talks. To sign up for this seminar series and receive zoom links via email, please follow the link. If you experience any difficulties with your registration, you may contact Natalia.Kopteva@ul.ie.
Thu 21 January 2021, 12:00 Noon GMT
Patrick Farrell (University of Oxford)
Reynolds-robust preconditioners for the stationary incompressible Navier-Stokes equations
When approximating PDEs with the finite element method, large sparse linear
systems must be solved. The ideal preconditioner yields convergence that is
algorithmically optimal and parameter robust, i.e. the number of Krylov
iterations required to solve the linear system to a given accuracy does not grow
substantially as the mesh or problem parameters are changed.
Achieving this for the stationary Navier-Stokes has proven challenging: LU
factorisation is Reynolds-robust but scales poorly with degree of freedom count,
while Schur complement approximations such as PCD and LSC degrade as the
Reynolds number is increased.
Building on the work of Schöberl, Olshanskii, and Benzi, in this talk we present
the first preconditioner for the Newton linearisation of the stationary
Navier--Stokes equations in three dimensions that achieves both optimal
complexity and Reynolds-robustness. The scheme combines augmented Lagrangian
stabilisation, a custom multigrid prolongation operator involving local solves
on coarse cells, and an additive patchwise relaxation on each level that
captures the kernel of the divergence operator.
We present 3D simulations with over one billion degrees of freedom with robust
performance from Reynolds number 10 to 5000. We also present recent extensions
to implicitly-constituted non-Newtonian problems, and to magnetohydrodynamics.
Thu 28 January 2021, 15:00 GMT
Abner Salgado (University of Tennessee)
Numerical methods for spectral fractional diffusion
We present and analyze finite element methods (FEMs) for the numerical approximation of the spectral fractional Laplacian. This method hinges on the extension to an infinite cylinder in one more dimension. We discuss rather delicate numerical issues that arise in the construction of reliable FEMs and in the a priori and a posteriori error analyses of such FEMs for both steady, and evolution fractional diffusion, both linear and nonlinear. We show illustrative simulations, applications, and mention challenging open questions.
- Thu 11 February 2021, 12:00 Noon GMT Gabriel Barrenechea (University of Strathclyde)
- Thu 25 February 2021, 12:00 Noon GMT Bangti Jin (University College London)
- Thu 11 March 2021, 12:00 Noon GMT Ivan Graham (University of Bath)
- Thu 25 March 2021, 12:00 Noon GMT Gunar Matthies (Technical University Dresden)