# INAF

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 started 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.

If you experience any difficulties with your registration, you may contact Natalia.Kopteva@ul.ie.

All seminar times are given in Dublin time; to convert them to your local time you may, for example, use the following Time Zone Converter.

## Forthcoming Seminars

• Thu 29 July 2021, 14:00 (Dublin)
Kirk Soodhalter (Trinity College Dublin)
• Thu 12 August 2021, 14:00 (Dublin)
• Thu 9 September 2021, 14:00 (Dublin)
Scott MacLachlan (Memorial University of Newfoundland)
• Thu 23 September 2021, 14:00 (Dublin)
Thomas Apel (Universität der Bundeswehr München)
• Thu 7 October 2021, 15:00 (Dublin)
Alan Demlow (Texas A&M University)

• Date + time t.b. confirmed
Natalia Kopteva (University of Limerick)

## Past Seminars

• Thu 1 July 2021
Erin Carson (Charles University in Prague)
The cost of iterative computations at scale
With exascale-level computation on the horizon, the art of predicting the cost of computations has acquired a renewed focus. This task is especially challenging in the case of iterative methods, for which a realistic convergence rate often cannot be determined with certainty a priori (unless we are satisfied with potentially outrageous overestimates) and which typically suffer from performance bottlenecks at scale due to synchronization cost. Moreover, the amplification of rounding errors can substantially affect the practical performance, in particular for methods with short recurrences.
In this talk, we focus on what we consider to be key points which are crucial to understanding the cost of iteratively solving linear algebraic systems, particularly in the context of Krylov subspace methods and their communication-avoiding variants. We argue that achieving optimal performance in practice will require a holistic approach, involving collaboration between the fields of numerical analysis, computer science, and computational sciences.
Slides

• Thu 17 June 2021
Kai Diethelm (University of Applied Sciences Würzburg-Schweinfurt)
Numerical Methods for Terminal Value Problems of Fractional Order
Traditionally, ordinary differential equations of fractional order $\alpha \in (0,1)$ are considered in combination with initial conditions, i.e.\ one imposes a condition on the unknown function at the starting point $a$, say, of the fractional differential operator in question. In practical applications, however, it is not always possible to provide the information about the unknown solution at this particular point. Rather, one is sometimes forced to use a condition of a form like $y(b) = y^*$ with some $b > a$. We briefly discuss analytic properties of such problems, in particular the questions of existence and uniqueness of their solutions. The main part of the talk will then be devoted to numerical methods for obtaining approximate solutions to problems of this type.
Slides

• Thu 3 June 2021
Bosco Garcia-Archilla (University of Seville)
Stabilized Finite Element Methods for the Navier-Stokes Equations
This talk will be a journey for non experts on the error analysis of finite element discretizations of the Navier-Stokes equations. We will start with the error analysis of the heat equation, and, step by step, we will add convection, compressibility and nonlinearity until reaching the Navier-Stokes equations. On each of these steps, we will analyse the effect of small diffusion on the error bounds, paying special attention to the reduction in order of convergence. Also we will comment on how the stabilization terms (terms added to the discretization improve the approximation) counterbalance the effect of small diffusion. Numerical examples will illustrate the different elements of the analysis.
Slides

• Thu 20 May 2021
Zhimin Zhang (Beijing Computational Science Research Center)
Superconvergence: An Old Field with New Territories
The phenomenon of superconvergence is well understood for the h-version finite element method, and researchers in this old field have accumulated a vast literature during the past 50 years. However, a similar study for other numerical methods such as the p-version finite element method, spectral methods, discontinuous Galerkin methods, and finite volume methods is lacking. We believe that the scientific community would also benefit from the study of superconvergence phenomena for those methods. In recent years, some efforts have been made to expand the territory of superconvergence analysis. In this talk, we present some recent developments in superconvergence analysis for discontinuous Galerkin methods and polynomial spectral methods. At the same time, some current issues and unsolved problems will also be addressed.
Slides

• Thu 6 May 2021
Catherine Powell (University of Manchester)
Adaptive Stochastic Galerkin Approximation for Parameter-Dependent PDEs
In this talk, we discuss numerical analysis aspects of stochastic Galerkin approximation for performing forward uncertainty quantification (UQ) in PDE models with uncertain (or parameter-dependent) inputs. Starting with a scalar elliptic test problem, we first describe a general strategy for performing a posteriori error estimation to drive adaptive solution algorithms. We then discuss how this methodology can be extended to a more challenging linear elasticity problem with uncertain Young’s modulus. We introduce a three-field parameter-dependent PDE model and develop an adaptive stochastic Galerkin mixed finite element scheme. We estimate the error in the natural weighted norm with respect to which the weak formulation is stable. Exploiting the connection between this norm and the underlying PDE operator also leads to an efficient block-diagonal preconditioning scheme for the associated discrete problems. Both the error estimator and the preconditioner are provably robust in the incompressible limit. If time allows, we will also discuss recent work for poroelasticity problems.
Slides

• Thu 22 April 2021
Ricardo Durán (University of Buenos Aires)
The Stokes equations with singular boundary data
First we recall some classic results on the well posedness and numerical approximation of the Stokes equations, particularly we present the fundamental inf-sup condition and the Bogovskii's constructive approach to prove it. Usually the theory is presented for the homogeneous Dirichlet problem but, by standard trace results, it can be extended to treat non-homogeneous boundary data provided they are enough regular.
Then, we consider the Dirichlet problem with singular data and analyze its finite element approximation. We prove quasi-optimal error estimates for data in fractional order Sobolev spaces approximating the boundary datum by appropriate regularizations, or by the Lagrange interpolation when it is piece-wise smooth.
A typical example used to test numerical methods is the so called lid-driven cavity problem. Our general results give almost optimal order error estimates for this case when quasi-uniform meshes are used.
Finally we comment on an a posteriori error estimator and present some numerical examples showing the good performance of an adaptive procedure based on it.
Slides

• Thu 8 April 2021
Emmanuil (Manolis) Georgoulis (University of Leicester/National Technical University of Athens)
hp-Version discontinuous Galerkin methods on arbitrarily-shaped elements
We extend the applicability of the popular interior-penalty discontinuous Galerkin (dG) method discretizing advection-diffusion-reaction problems to meshes comprising extremely general, essentially arbitrarily-shaped element shapes. In particular, our analysis allows for curved element shapes, without the use of nonlinear elemental maps. The feasibility of the method relies on the definition of a suitable choice of the discontinuity penalization, which turns out to be explicitly dependent on the particular element shape, but essentially independent on small shape variations. This is achieved upon proving extensions of classical inverse estimates to arbitrary element shapes. These inverse estimates may be of independent interest. A priori error bounds for the resulting method are given under very mild structural assumptions restricting the magnitude of the local curvature of element boundaries. We further discuss the applicability of this new framework within adaptive algorithms and discuss briefly the proof of a posteriori error bounds.
Slides

• Thu 25 March 2021
Gunar Matthies (Technical University Dresden)
Local projection stabilisation
Originally proposed to stabilise equal-order discretisations of the Stokes problem, local projection stabilisation (LPS) has been applied successfully stabilise dominating convection in both convection-diffusion equations and incompressible flow problems.
The first part will consider convection-diffusion equations and discuss the role of a special interpolation operator that is used in the numerical analysis. We will give conditions that ensure its existence and present some example settings. Numerical results will illustrate the behaviour of local projection stabilsation.
The second part of the talk will present some results for Oseen problems where we consider both equal-order and inf-sup stable discretisations. We will give also some numerical results.
Slides

• Thu 11 March 2021
Ivan Graham (University of Bath)
Solving the Helmholtz equation at high frequency
The Helmholtz equation arises when the linear wave equation is reduced to a steady state PDE via Fourier transform in time. It is arguably one of the simplest equations describing linear waves in general geometries and media, and it provides a scalar model for more complicated problems like the elastic wave equation or Maxwell's equations. It arises in many applications, including inverse problems e.g., seismic imaging. Despite it's linearity and apparent simplicity, this equation is difficult to solve because (a) its stability properties are complicated and depend on domain geometry and material properties of the medium; (b) at high frequency, solutions are highly oscillatory, very fine meshes are needed to even guarantee the existence/uniqueness of numerical solutions, and finer meshes are needed for accuracy; (c) the system matrices which arise after discretization are highly indefinite and non-normal, and (in contrast to positive definite PDE problems), the formulation and analysis of fast parallel iterative methods is difficult. On the last point, there is currently intense research interest amongst a number of groups worldwide on developing efficient linear solvers.
In the talk I'll give some background to the Helmholtz problem, describe what is known about its stability and finite element error analysis and then I'll describe work I have been doing with colleagues on the formulation and analysis of domain decomposition methods for solving the linear systems arising from discretized Helmholtz problems. My main collaborators for the talk are Shihua Gong and Euan Spence (both of Bath) and Jun Zou (Chinese University of Hong Kong), although other collaborators will also be mentioned during the talk.

Slides

• Thu 25 February 2021
Bangti Jin (University College London)
Numerical methods for time-fractional diffusion
During the past decade, parabolic type equations involving a fractional-order derivative in time have received much attention, and several numerical methods have been developed. Many existing methods are developed by assuming that the solution is sufficiently smooth. In this talk, I will describe some works on developing robust numerical schemes that do not assume solution regularity directly, but only data regularity.
Slides

• Thu 11 February 2021
Gabriel Barrenechea (University of Strathclyde)
The discrete maximum principle in finite element methods
In this talk the satisfaction of the discrete maximum principle for the finite element method will be reviewed. Starting from the most basic results on the topic, and basing ourselves in the algebraic equations, sufficient conditions for the satisfaction of the discrete maximum principle for nonlinear discretisations (of shock-capturing kind) will be given. As an example of such discretisations the family of algebraic flux correction schemes will be analysed in the case of the convection-diffusion equation, where the role of the mesh geometry will be studied.
Slides

• Thu 28 January 2021
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.

Slides

• Thu 21 January 2021
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.

Funding and support:
There is no registration fee for the workshop. The organisers are grateful to the Irish Mathematical Society, for their generous financial support.

Organisers: Alan Hegarty, Natalia Kopteva, Niall Madden, Eugene O'Riordan, Kirk Soodhalter and Martin Stynes.