New Trends in the Numerical Analysis of PDEs

Contribution List

1. Structure-preserving discretization of nonlinear cross-diffusion systems

Ilaria Perugia (Universität Wien)

10/06/2024, 14:30

The main challenges in designing numerical methods for approximating nonlinear cross-diffusion systems is that the diffusion matrix may not be symmetric or positive semidefinite, and that a maximum principle may be not available. In this talk, we present a Local Discontinuous Galerkin method for discretizing nonlinear cross-diffusion systems, which is based on the boundedness-by-entropy...

4. Computable reliable bounds for Poincaré–Friedrichs constants via Čech–de-Rham complexes

Martin W. Licht (EPFL)

10/06/2024, 15:15

We derive computable and reliable upper bounds for Poincaré–Friedrichs constants of classical Sobolev spaces and, more generally, Sobolev de-Rham complexes. The upper bounds are in terms of local Poincaré–Friedrichs constants over subdomains and the smallest singular value of a finite-dimensional operator that is easily assembled from the geometric setting. Thus we reduce the computational...

3. A posteriori goal-oriented error estimators based on equilibrated flux and potential reconstructions

Emmanuel Creusé (Université Polytechnique Hauts-de-France)

10/06/2024, 16:30

Nowadays, many engineering problems require computing some quantities of interest, which are usually linear functionals applied to the solution of a partial differential equation. Error estimations of such functionals are called "goal-oriented" error estimations. Such estimations are based on the resolution of an adjoint problem, whose solution is used in the estimator definition, and on the...

2. Basics for polynomial interpolation on simplices

Francesca Rapetti (Université Côte d'Azur)

10/06/2024, 17:15

Polynomial interpolation is a key aspect in numerical analysis, used in very classical settings as for reconstructing a field from measures, computing integrals by quadratures formulas or selecting basis functions in finite element methods. We will review the roles of the three main characters featured in this action, namely, the representation of the domain by a mesh, the polynomial basis,...

5. Hybrid compatible Finite Element and Finite Volume discretization for viscous and resistive MHD

Eric Sonnendrücker (Max Planck Institute for Plasma Physics and TU München)

11/06/2024, 09:30

MHD simulations including small viscous and resistive effects are fundamental for simulations related to magnetic fusion.
However, due to the needed long time simulations and the very different wave speeds, implicit or semi-implicit methods are unavoidable.
On the other hand, div B = 0 as well as other symmetries and invariants need to be preserved by the numerical algorithm.
To this aim,...

6. The Exterior Calculus Discrete De Rham complex

Jérôme Droniou (CNRS, Université de Montpellier and Monash University)

11/06/2024, 10:45

In its standard presentation, the de Rham complex organises the gradient, curl and divergence operator into a sequence that embeds the well-known calculus relations: the image of one operator (e.g. gradient) is included in the kernel of the following one (e.g. curl). The de Rham theorem states that the gaps between these images and kernels, embedded into the cohomology of the complex, is...

7. Polytopal methods on Riemannian manifolds

Marien-Lorenzo Hanot (The University of Edinburgh)

11/06/2024, 11:30

Discretization methods based on differential complexes have many advantageous properties in terms of stability, framework for analysing the discrete formulation, and preservation of important quantities such as the mass, helicity, or the pressure robustness in fluid dynamics.

The premise of this kind of approach appeared early on with elements based on the compatibility between the geometry...

9. Learning based reduction methods in the context of PDE constrained optimization

Mario Ohlberger (Universität Münster)

11/06/2024, 14:30

Model order reduction for parameterized partial differential equations is a very active research area that has seen tremendous development in recent years from both theoretical and application perspectives. A particular promising approach is the reduced basis method that relies on the approximation of the solution manifold of a parameterized system by tailored low dimensional approximation...

10. Global space-time low-rank methods for the time-dependent Schrödinger equations

Virginie Ehrlacher (Ecole des Ponts ParisTech and Inria Paris Centre)

11/06/2024, 15:15

The aim of this talk is to present novel global space-time methods for the approximation of the time-dependent Schrödinger equation, using Kato theory. The latter can be used in conjunction with low-rank tensor formats (such as Tensor Trains for instance) to derive new variational principles to compute dynamical low-rank approximations of the solution, which are different from the...

21. Runge-Kutta methods are stable

Eitan Tadmor (University of Maryland)

11/06/2024, 16:00

The numerical solution of PDEs often ends up with a large system of ODEs, and a canonical choice for the solution of such systems of “method of lines” is the class of Runge-Kutta (RK) methods. Indeed, RK methods are used routinely for integration of large systems of ODEs encountered in various applications. But the standard stability arguments of RK method fail to cover arbitrarily large...

8. Poster session

11/06/2024, 17:15

Margherita Castellano (Ecole Polytechnique): A finite volume method for Cahn--Hilliard equations with surfactants

Jean Cauvin-Vila (TU Vienna): Structure-preserving finite volume approximation of cross-diffusion systems coupled via a moving interface

Farah Chaaban (ENSTA Paris): A volume optimal control-based numerical method for 2D time-harmonic Maxwell's equations with sign-changing...

11. Modeling Multiphase Multicomponent Porous Flows

Noel J. Walkington (Carnegie Mellon University)

12/06/2024, 09:30

This talk will review structural properties of the equations used to model porous flows involving multiple components undergoing phase transitions. These equations only model the gross properties of these problems since a precise description of the physical system is neither available nor computationally tractable. The saddle point structure resulting from the interaction between dissipation...

12. Asymptotic analysis of a fluid-structure-porous layer coupled model near contact

Céline Grandmont (Inria Paris Centre and Université Libre de Bruxelles)

12/06/2024, 10:45

The numerical simulation of systems involving fluid-structure-contact interaction raises many modeling, mathematical and numerical difficulties. It is in particular crucial for numerous biomedical applications such as the simulation of cardiac valve dynamics (native or artificial) for instance. Fluid-structure interaction without contact is already challenging due to the moving geometries and...

13. A nonlinear reduced model based on optimal transport for electronic structure calculations

Geneviève Dusson (CNRS, Université Bourgogne Franche-Comté)

12/06/2024, 11:30

Electronic structure calculations are widely used to predict the physical properties of molecules and materials. They require to solve nonlinear partial differential and eigenvalue equations. These equations are generally numerically very demanding, especially since they are parameterized by the positions of the nuclei in the molecule and must be solved a large number of times when these...

14. On the approximation of the von Neumann equation in the semi-classical limit

Francis Filbet (Université Paul Sabatier)

12/06/2024, 14:30

We propose a new approach to discretize the von Neumann equation, which is efficient in the semi-classical limit. This method is first based on the so called Weyl’s variables to address the stiffness associated with the equation. Then, by applying a truncated Hermite expansion of the density operator, we successfully handle this stiffness. Additionally, we develop a finite volume approximation...

15. Neural and hybrid methods for elliptic problems

Emmanuel Franck (Inria Centre at Université de Lorraine)

12/06/2024, 15:15

First, we'll introduce the neural methods used to solve PDEs, such as PINNs or the Deep Ritz method. It will be shown that these approaches can fit within the framework of classical Galerkin methods, where only the approximation space changes. The advantages and shortcomings of these approaches will be discussed. Next, a "prediction-correction" approach will be proposed, in which neural...

16. A multi-dimensional staggered scheme for the diffusive limit in the radiative transfer equation

Khaled Saleh (Université Claude Bernard, Lyon 1)

12/06/2024, 16:30

The radiative transfer equation is a kinetic PDE modelling the specific radiation intensity carried by a population of photons described by a statistical description, i.e. a transport equation on the fraction of photons travelling in a given direction. It is well known that as the Knudsen number (which is the ratio of the mean free path length to a representative physical length scale) goes to...

17. A structure-preserving semi-implicit IMEX finite volume scheme for ideal magnetohydrodynamics at all Mach and Alfvén numbers

Andrea Thomann (Inria Centre at Université de Lorraine)

12/06/2024, 17:15

We present a divergence-free semi-implicit finite volume scheme for the simulation of the ideal magnetohydrodynamics (MHD) equations which is stable for large time steps controlled by the local transport speed at all Mach and Alfvén numbers. An operator splitting technique allows to treat the convective terms explicitly while the hydrodynamic pressure and the magnetic field contributions are...

18. Designing conservative and accurately dissipative numerical integrators in time

Patrick E. Farrell (University of Oxford)

13/06/2024, 09:30

Numerical methods for the simulation of transient systems with structure-preserving properties are known to exhibit greater accuracy and physical reliability, in particular over long durations. These schemes are often built on powerful geometric ideas for broad classes of problems, such as Hamiltonian or reversible systems. However, there remain difficulties in devising higher-order- in-time...

19. A posteriori error control in the max norm for the Monge-Ampère equation

Dietmar Gallistl (Universität Jena)

13/06/2024, 10:45

This talk discusses a stability result for the Monge-Ampère operator in a (potentially regularized) Hamilton-Jacobi-Bellman format as a consequence of Alexandrov's classical maximum principle. The main application is guaranteed a posteriori error control in the $L^\infty$ norm for the difference of the Monge-Ampère solution and the convex hull of a fairly arbitrary $C^1$-conforming finite...

20. Analysis and numerical approximation of mean field game partial differential inclusions

Iain Smears (University College London)

13/06/2024, 11:30

Joint work with Yohance A. P. Osborne

Mean field games (MFG) are models for differential games involving large numbers of players, where each player is solving a dynamic optimal control problem that may depend on the overall distribution of players across the state space of the game. In a standard formulation, the Nash equilibria of the game are characterized by the solutions of a coupled...