## 2023 Seminars

**Date: **Thursday, 16 March 2023, 14:00 (Dublin)

David Cohen (Chalmers University of Technology, Sweden)

**Numerical analysis of stochastic Poisson systems**

The subject of the presentation is the study of splitting integrators for Poisson systems perturbed by additive noise and multiplicative Stratonovich noise.

The presentation is based on a joint work with G. Vilmart and on a joint work with C-E. Bréhier and T. Jahnke.

**Slides**

Thu 23 February 2023

Wil Schilders (Eindhoven University of Technology, The Netherlands, & TU Munich – Institute of Advanced Study, Germany)

**Mathematics: key enabling technology for scientific machine learning **

Artificial Intelligence (AI) will strongly determine our future prosperity and well-being. Due to its generic nature, AI will have an impact on all sciences and business sectors, our private lives and society as a whole. AI is pre-eminently a multidisciplinary technology that connects scientists from a wide variety of research areas, from behavioural science and ethics to mathematics and computer science.

Without downplaying the importance of that variety, it is apparent that mathematics can and should play an active role. All the more so as, alongside the successes of AI, also critical voices are increasingly heard. As Robert Dijkgraaf (former director of the Princeton Institute of Advanced Studies) observed in May 2019: ”Artificial intelligence is in its adolescent phase, characterised by trial and error, self-aggrandisement, credulity and lack of systematic understanding.” Mathematics can contribute to the much-needed systematic understanding of AI, for example, greatly improving reliability and robustness of AI algorithms, understanding the operation and sensitivity of networks, reducing the need for abundant data sets, or incorporating physical properties into neural networks needed for superfast and accurate simulations in the context of digital twinning.

Mathematicians absolutely recognise the potential of artificial intelligence, machine learning and (deep) neural networks for future developments in science, technology and industry. At the same time, a sound mathematical treatment is essential for all aspects of artificial intelligence, including imaging, speech recognition, analysis of texts or autonomous driving, implying it is essential to involve mathematicians in all these areas. In this presentation, we highlight the role of mathematics as a key enabling technology within the emerging field of scientific machine learning. Or, as I always say: ”Real intelligence is needed to make artificial intelligence work.”

**Slides**

Thu 9 February 2023

Conall Kelly (University College Cork, Ireland)

**An adaptive splitting method for the Cox-Ingersoll-Ross process**

The Cox-Ingersoll-Ross (CIR) process is described by an It\^o-type stochastic differential equation with a square-root diffusion and appears frequently in applications such as finance and neuroscience. In finance CIR is used as a model of interest rates or stochastic volatility. Solutions are almost surely (a.s.) non-negative; in fact under a parameter constraint called Feller's condition, they are known to be a.s. positive.

The challenge for any numerical method applied to an SDE with square-root diffusion is to control error induced by the discretisation in spite of the unbounded gradient of the square-root coefficient near zero, and to preserve the a.s. positivity of trajectories. CIR is a useful test equation in this regard since techniques developed to handle the functional response near zero can be applied to more general equations.

We propose a domain invariant numerical method for CIR applied over both deterministic and adaptive random meshes and based upon a suitable transform followed by a splitting. Moment bounds and theoretical strong $L_2$ and $L_1$ convergence rates of order $1/4$ the scheme are available in a restricted parameter regime. We then extend the new method to cover all parameter values by introducing a soft zero region (where the deterministic flow determines the approximation) resulting in a hybrid method that deals with the reflecting boundary.

From numerical simulations we observe an optimal convergence rate of $1$ within the Feller regime. As the stochastic intensity increases and we move outside of this parameter region, we observe that the rates of strong convergence are competitive with other schemes in terms of convergence order, however the proposed method with adaptive timestepping consistently displays smaller error constants.

**Slides**

Thu 26 January 2023

Alex Bespalov (University of Birmingham, UK)

**Multilevel adaptivity for stochastic finite element methods**

This talk is concerned with the design and analysis of adaptive FEM-based solution strategies for partial differential equations (PDEs) with uncertain or parameter-dependent inputs. We consider a projection-based method (stochastic Galerkin FEM) and a sampling-based method (stochastic collocation FEM). Both strategies have emerged and become popular as effective alternatives to Monte-Carlo sampling in the context of (forward) uncertainty quantification. In both stochastic Galerkin and stochastic collocation methods, the approximations are represented as finite (sparse) expansions in terms of a parametric polynomial basis with spatial coefficients residing in finite element spaces. The focus of the talk is on multilevel approaches where different spatial coefficients may reside in different finite element spaces and, therefore, the underlying spatial approximations are allowed to be refined independently from each other.

We start with a more familiar setting of projection-based methods, where exploiting the Galerkin orthogonality property and polynomial approximations in terms of an orthonormal basis facilitates the design and analysis of adaptive algorithms. We discuss a posteriori error estimation as well as the convergence and rate optimality properties of the generated adaptive multilevel Galerkin solutions for PDE problems with affine-parametric coefficients. We then show how these ideas of error estimation and multilevel adaptivity can be applied in a non-Galerkin setting of stochastic collocation FEM, in particular, for PDE problems with non-affine parameterization of random inputs and for problems with parameter-dependent local spatial features.

The talk is based on a series of joint papers with Dirk Praetorius (TU Vienna), Leonardo Rocchi (Birmingham), Michele Ruggeri (University of Strathclyde, Glasgow), David Silvester (Manchester), and Feng Xu (Manchester).

**Slides**

## 2022 Seminars

12 May 2022

Alexandre Ern (ENPC and INRIA Paris, France)

**Hybrid high-order methods for the biharmonic problem**

We start with a gentle introduction to the devising and analysis of hybrid high-order (HHO) methods for the Poisson model problem. Then, we address the biharmonic problem and we compare the proposed HHO methods to the literature, in particular to weak Galerkin methods. Finally, we briefly discuss how the error analysis can be carried out in the case of an exact solution with low regularity.

**Slides**

Thu 28 April 2022

Endre Süli (University of Oxford, UK)

**Finite element approximation of nonlinear elliptic systems with linear growth**

The talk is concerned with the numerical analysis of a system of nonlinear partial differential equations that arises from Rajagopal's theory of elastic solids with limiting small strain. We construct a finite element approximation of this limited-strain elastic model. The sequence of finite element approximations is shown to exhibit strong convergence to the unique weak solution of the model. Assuming that the material parameters featuring in the model are Lipschitz-continuous, and assuming that the weak solution has additional regularity, the sequence of approximations is shown to converge with a rate. The theoretical results are illustrated by numerical experiments.

The talk is based on a series of joint papers with Lisa Beck (Augsburg), Andrea Bonito (Texas A&M), Miroslav Bulicek (Charles University Prague), Vivette Girault (Paris VI), Josef Malek (Charles University Prague), and Kumbakonam Rajagopal (Texas A&M).

**Slides**

Thu 14 April 2022

Michael Tretyakov (University of Nottingham, UK)

**Simplest random walks for boundary value problems**

Solutions of boundary value problems for linear parabolic and elliptic PDEs can be represented as expectations of solutions of the corresponding systems of SDEs in bounded domains. Approximations of these systems of SDEs should be subject to restrictions related to their nonexit from the bounded domains. In the talk an overview of simple weak methods for SDEs suitable for solving the Dirichlet and Robin boundary value problem for parabolic and elliptic PDEs will be given. An approximation of SDEs driven by Lévy processes related to solving the Dirichlet boundary value problem for a linear parabolic PIDE will be also discussed.

**Slides**

Thu **7 April** 2022 (please NOTE the date change!)

Pavel Bochev (Sandia National Laboratories, USA)

**Data-driven, compact models for radiation-induced photocurrent effects**

When a semiconductor device is exposed to a pulse of ionizing radiation, it produces excess carriers that are not present in normal environments. The photocurrents generated by these carriers can alter the behavior of the device and present threats for microelectronic components operating in radiation environments such as navigation and communication satellites. Analysis of radiation effects on electrical circuits requires computationally efficient compact radiation models. Currently, development of such models is dominated by analytic techniques that rely on empirical assumptions and physical approximations to render the governing equations solvable in closed form.

At the same time, data-driven techniques combining system identification and/or model order reduction (MOR) have been successfully applied across many disciplines. However, despite their proven ability to produce computationally efficient reduced order models (ROM), these ideas have not yet been explored for radiation effects, where traditional analytic techniques dominate. In this context, data-driven approaches hold a significant promise as a compact model development tool because they have the potential to deliver compact models unburdened by the physical approximations and assumptions necessary for the analytic models.

In this talk we present a hybrid analytic-numerical approach to replace analytic solutions of the governing equations by numerical ones obtained from synthetic and/or experimental data by using a hierarchy of data-driven and machine-learning approaches, combined with model-order-reduction techniques. This obviates the need for additional approximations and yields a hierarchy of accurate and computationally efficient compact photocurrent models. We demonstrate these models by comparing their predictions with those of state-of-the-art analytic models using synthetic data and photocurrent measurements obtained at the Little Mountain Test Facility at Hill AFB, Utah.

**Slides**

Thu 10 March 2022

Chris Budd (University of Bath, UK)

**r-adaptivity, deep learning and the deep Ritz method**

Physics informed neural networks (PINNS) have recently become popular as an alternative way of solving differential equations in a 'mesh free manner'. In this talk I will describe the operation of both PINNS and variational forms of PINNS. I will demonstrate, by looking at some challenging problems, that despite their claim to be mesh free methods, a careful use of a computational mesh can dramatically improve their importance. This gives some insight into the way that numerical analysis and machine learning can work well together.

Joint work with Simone Appella (PhD, Bath) and Tristan Pryer (Bath)

**Slides**

Thu 24 February 2022

Qiang Du, (Columbia University, USA)

**Linear Multistep Methods for Learning Unknown Dynamics**

Numerical integration of a given dynamic system can be viewed as a forward problem with the learning of unknown dynamics from available state observations as an inverse problem. The latter has been around in various settings such as model reductions of multiscale processes. It has received particular attention recently in the data-driven modeling via deep/machine learning. The solution of both forward and inverse problems forms the loop of informative and intelligent scientific computing. A related question is whether a good numerical integrator for discretizing prescribed dynamics is also good for discovering unknown dynamics, in association with deep learning. This lecture presents a study in the context of linear multistep methods, revealing a less studied aspect of this classical numerical analysis subject. The talk should be accessible to people with very basic knowledge of numerical methods.

**Slides**

Thu 10 February 2022

Johnny Guzman (Brown University, USA)

**Local and L^2 bounded projections in FEEC**

One of the chief tools of Finite element exterior calculus (FEEC) are bounded commuting projections. They provided stability and convergence results for FEEC applied to the Hodge Laplacian. Early in the development of FEEC such projections were constructed by Schoberl and Christiansnen and Winther. More recently Falk and Winther have developed projections that are also local. Inspired by these results we discuss new projections that are local and bounded in L^2.

This is joint work with Douglas Arnold.

Thu 27 January 2022

Evelyn Buckwar (Johannes Kepler University Linz, Austria)

**Splitting methods in Approximate Bayesian Computation for partially observed diffusion processes**

Approximate Bayesian Computation (ABC) has become one of the major tools of likelihood-free statistical inference in complex mathematical models. Simultaneously, stochastic differential equations (SDEs) have developed as an established tool for modelling time dependent, real world phenomena with underlying random effects. When applying ABC to stochastic models, two major difficulties arise. First, the derivation of effective summary statistics and proper distances is particularly challenging, since simulations from the stochastic process under the same parameter configuration result in different trajectories. Second, exact simulation schemes to generate trajectories from the stochastic model are rarely available, requiring the derivation of suitable numerical methods for the synthetic data generation. In this talk we consider SDEs having an invariant density and apply measure-preserving splitting schemes for the synthetic data generation. We illustrate the results of the parameter estimation with the corresponding ABC algorithm with simulated data.This talk is based on joined work with Massimiliano Tamborrino, University of Warwick, and Irene Tubikanec, Johannes Kepler University Linz.

**Slides**

Thu 13 January 2022

Georgios Akrivis (Institute of Applied and Computational Mathematics, FORTH, Heraklion, Crete, Greece)

**The energy technique for BDF methods**

The energy technique (or method) is probably the easiest way to establish stability of parabolic (and other) differential equations.

The application of the energy technique to numerical methods with very good stability properties, such as algebraically stable Runge-Kutta methods or A-stable multistep methods, is straightforward. The extension to other numerical methods, such as A($\vartheta$)-stable methods, requires some effort and is more interesting; the main difficulty concerns suitable choices of test functions.

Here we focus on the energy technique for backward difference formula (BDF) methods. In the cases of the A-stable one- and two-step BDF methods the application is trivial. The energy technique is applicable also to the A($\vartheta$)-stable three-, four- and five-step BDF methods via Nevanlinna--Odeh multipliers.

The main results are:

+ No Nevanlinna--Odeh multipliers exist for the six-step BDF method.

+ The energy technique is applicable under a relaxed condition on the multipliers.

+ We present multipliers that make the energy technique applicable also to the six-step BDF method.

Besides its simplicity, the energy technique is *powerful*, it leads to several stability estimates, and *flexible*, it can be easily combined with other stability techniques.

The talk is based on the paper:

G. A., M. Chen, F. Yu, Z. Zhou: *The energy technique for the six-step BDF method*, SIAM J. Numer. Anal., 59 (2021), 2449-2472.

Joint work with:

Minghua Chen, Fan Yu: School of Mathematics and Statistics, Gansu Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou, P.R. China;

Zhi Zhou: Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong.

**Slides** + **Supplement**

## 2021 Seminars

Thu 2 December 2021

Gabriel Lord (Radboud University, Nijmegen)

**Adaptive timestepping for S(P)DEs**

Traditional explicit numerical methods to simulate stochastic differential equations (SDEs) or stochastic partial differential equations (SPDEs) rely on globally Lipschitz coefficients to ensure convergence. Many applications of interest include non Lipschitz drift functions. Implicit methods (when they exist) can often be too computationally expensive for practical uses and standard explicit methods suffer from growth of moments of the solution (which can be thought of as a form of numerical instability).

Therefore the construction of explicit methods to simulate SDEs or SPDEs with non-Lipschitz drift has been an area of great interest. These methods are broadly in a number of classes where either the numerical approximations are projected or the growth is controlled in the scheme.

In this talk we give an overview, emphasizing the context of SPDEs, of this issue and discuss how using an adaptive time step can be used to control this growth. We will discuss some of the key difficulties in proving strong convergence with a random time mesh and how these might be overcome. We show that in numerical experiments the adaptive time stepping is an efficient alternative to the other methods.

This work is joint with Conall Kelly (UCC) and Fandi Sun and Stuart Campbell (Heriot-Watt University)

**Slides**

Andy Wathen (University of Oxford)

**Parallel preconditioning for time-dependent PDEs**

Large scale simulations with partial differential equations (PDEs) demand significant computational resources---it is one of the problem areas where parallel computation is often a necessity. Ideas for parallel solution of stationary PDE problems have been widely explored, with Domain Decomposition preconditioning being a leading technique. For evolutionary PDEs, however, it would seem that causality---the need for the solution at earlier time in order to realise it later---should severely limit possibilities for effective parallel computation over time.

In this talk we develop one possible `parallel-in-time' approach that involves ideas and algorithms from numerical linear algebra.

Kirk Soodhalter (Trinity College Dublin)

**Analysis of block GMRES using a new *-algebra-based approach**

In this talk, we give an general overview of a class of iterative methods for solving linear systems called Krylov subspace methods (KSM) and their block generalizations, focusing on the Generalized Minimum RESidual method (GMRES). We then discuss the challenges of extending convergence results of classical GMRES to its block counterpart and propose a new approach to this analysis. Block KSMs such as block GMRES are generalizations of classical KSMs, and are meant to iteratively solve linear systems with multiple right-hand sides (a.k.a. a block right-hand side) all-at-once rather than individually. These methods have regained attention recently due in part to the computational advantages they offer in the high-performance computing setting. However, this all-at-once approach has made analysis of these methods more difficult than for classical KSMs because of the interaction of the different right-hand sides. We have proposed an approach built on interpreting the coefficient matrix and block right-hand side as being a matrix and vector over a *-algebra of square matrices. This allows us to sequester the interactions between the right-hand sides into the elements of the *-algebra and extend some classical GMRES convergence results to the block setting. We then discuss some challenges which remain and some ideas for how to proceed.

**Slides**

Michal Križek (Czech Academy of Sciences, Prague)

**Finite element approximation of a nonlinear heat conduction problem in anisotropic media**

This lecture will be a survey of results which we have obtained in solving a stationary nonlinear heat condiction problem by the finite element method. In particular, we present uniqueness theorems for the classical and weak solutions, a comparison theorem, existence theorems for the weak and finite element solutions, convergence of finite element approximations without any regularity assumptions, a priori error estimates, numerical integration, discrete maximum principle, and nonlinear radiation boundary conditions.

**Slides**

Alan Demlow (Texas A&M University)

**Geometric errors in surface finite element methods**

The Laplace-Beltrami problem is widely used to model physical and other phenomena on surfaces. Finite element methods a widely used option to solve such equations. Typically in FEM the surface on which the PDE is posed is first approximated by a nearby discrete surface, and then the finite element method is posed on the discrete surface much as for PDE on Euclidean spaces. Replacing the original continuous surface by a discrete approximation gives rise to a consistency error (variational crime) that is typically called a geometric error. Until a few years ago, instances of geometric errors appearing in the literature exhibited consistent behavior across a range of situations. More recently a number of cases have arisen in which more subtle behavior occurs. We will describe a couple of such cases, those of eigenvalue problems and a posteriori error estimation.

**Slides**

Thomas Apel (Universität der Bundeswehr München)

**A pressure-robust discretization of the Stokes problem on anisotropic meshes**

(joint work with Volker Kempf)

Anisotropic finite element meshes are particularly efficient when the solution of the problem has anisotropic features like boundary layers or edge singularities. Pressure-robustness means that the velocity discretization error does not depend on the pressure approximation. The Crouzeix-Raviart method is known to be inf-sup stable on arbitrary anisotropic meshes, yet not pressure-robust.

Inf-sup stable finite element schemes with discontinuous pressure can be made pressure-robust by a modified discretization of the exterior forcing term using H(div)-conforming reconstruction operators like the Raviart-Thomas or Brezzi-Douglas-Marini interpolants, [1].

In order to show that the reconstruction approach works for anisotropic discretizations, it was necessary to investigate the interpolation error for the Raviart-Thomas or Brezzi-Douglas-Marini interpolants on anisotropic elements disclosing subtleties in the definition of the spaces and in shape assumptions, [2,3].

In collaboration with Alexander Linke and Christian Merdon we have generalized the modified Crouzeix-Raviart method to a large class of anisotropic triangulations. Numerical examples confirm the theoretical results, [4,5].

[1] A. Linke: On the role of the Helmholtz decomposition in mixed methods for incompressible flows and a new variational crime. Comput. Methods Appl. Mech. Engrg. 268(2014), 782-800.

[2] G. Acosta, Th. Apel, R. G. Durán, A. L. Lombardi: Error estimates for Raviart-Thomas interpolation of any order on anisotropic tetrahedra. Math. Comp. 80(2011), 141-163.

[3] Th. Apel, V. Kempf: Brezzi-Douglas-Marini interpolation of any order on anisotropic triangles and tetrahedra. SIAM J. Numer. Anal. 58(2020), 1696-1718.

[4] Th. Apel, V. Kempf, A. Linke, Chr. Merdon: A nonconforming pressure-robust finite element method for the Stokes equations on anisotropic meshes. IMA Journal of Numerical Analysis (2021).

[5] Th. Apel, V. Kempf: Pressure-robust error estimate of optimal order for the Stokes equations: domains with re-entrant edges and anisotropic mesh grading. Calcolo 58(2021).

**Slides**

Scott MacLachlan - (Memorial University of Newfoundland)

**Finite-Element Modeling of Liquid Crystal Equilibria**

Numerical simulation tools for fluid and solid mechanics are often based on the discretisation of coupled systems of partial differential equations, which can easily be identified in terms of physical conservation laws. In contrast, equilibrium configurations of many liquid crystal phases are more naturally described by the first-order optimality conditions of constrained free-energy functionals. In this talk, I will present a variational finite-element approach for computing liquid crystal equilibria, and demonstrate its use for both nematic (rod-like) and smectic (soap-like) liquid crystals. As the main scientific and engineering interest in liquid crystals comes from their ability to exhibit multiple distinct stable equilibrium states, I will discuss the combination of this framework with a nonlinear deflation technique that allows discovery of the energy landscapes for these problems.

**Slides** + **Videos**

Carmen Rodrigo (Universidad de Zaragoza)

**Robust discretizations and solvers for poroelastic problems**

The numerical simulation of poroelastic problems has received a lot of attention due to their wide range of applications. Geothermal energy extraction, CO2 storage, hydraulic fracturing or cancer research are among typical societal relevant applications of poromechanics. Robust discretizations with respect to all the physical parameters are needed for this type of problems to obtain reliable numerical solutions. This is a very important task and some efforts are being carried out in this address by the scientific community. In particular, we present here a stable discretization for the three field formulation of the poroelasticity problem. In addition, intensive research has also been focused on the design of efficient methods for solving the large linear systems arising from the discretization of Biot's model, since in real simulations it is the most consuming part. We aim to design efficient and robust preconditioners to accelerate the convergence of Krylov subspace methods. The proposed block preconditioners for solving the Biot's model are based on the well-posedness of the obtained discrete systems, and are robust with respect to both physical and discretization parameters. Numerical results are presented to support the theoretical results.

Slides

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.

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.

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**

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**

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**

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**

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**

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**

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.

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**

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**

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.

**Slides**

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.