New Trends in the Numerical Analysis of PDEs

10 Jun 2024, 12:30 → 13 Jun 2024, 14:00 Europe/Paris

Amphitheater, Building B (Inria Center at the University of Lille)

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The goal of this event is to gather experts in the numerical analysis of PDEs. The spectrum of the conference is deliberately broad, and open to different active branches of the domain, such as:

• Physical modeling with PDEs
• Design and analysis of structure-preserving numerical schemes
• High-order numerical methods on general meshes 
• A priori and a posteriori analysis of numerical methods
• Model reduction and high performance computing

 

Confirmed speakers

• Emmanuel Creusé (Université Polytechnique Hauts-de-France)
• Jérôme Droniou (CNRS, Université de Montpellier and Monash University)
• Geneviève Dusson (CNRS, Université Bourgogne Franche-Comté)
• Virginie Ehrlacher (Ecole des Ponts ParisTech and Inria Paris Centre)
• Patrick E. Farrell (University of Oxford)
• Francis Filbet (Université Paul Sabatier)
• Emmanuel Franck (Inria Centre at Université de Lorraine)
• Dietmar Gallistl (Universität Jena)
• Céline Grandmont (Inria Paris Centre and Université Libre de Bruxelles)
• Marien-Lorenzo Hanot (The University of Edinburgh)
• Martin W. Licht (EPFL)
• Mario Ohlberger (Universität Münster)
• Ilaria Perugia (Universität Wien)
• Francesca Rapetti (Université Côte d'Azur)
• Khaled Saleh (Université Claude Bernard, Lyon 1)
• Iain Smears (University College London)
• Eric Sonnendrücker (Max Planck Institute for Plasma Physics and TU München)
• Eitan Tadmor (Fondation Sciences Mathématiques de Paris (FSMP) and University of Maryland)
• Andrea Thomann (Inria Centre at Université de Lorraine)
• Noel J. Walkington (Carnegie Mellon University)

 

Practical information

Registration is free but mandatory. You can submit an abstract for the Tuesday's poster session. Limited funding is available for early stage researchers upon demand.

 

 

Monday, 10 June

12:30 → 14:20 Welcome buffet

14:20 → 14:30 Workshop opening

14:30 → 15:15 Structure-preserving discretization of nonlinear cross-diffusion systems

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 framework introduced by A. Jüngel in 2015. Motivated by the underlying entropy structure of the PDE system, nonlinear transformations in terms of the entropy variable allow to enforce positivity of approximate solutions. Moreover, by appropriately introducing auxiliary variables, the problem is reformulated so that nonlinearities do not appear within differential operators or interface terms, leading to nonlinear operators that can be naturally evaluated in parallel. The resulting method has the following desirable properties:

i) it allows arbitrary degrees of approximation in space;
ii) it preserves boundedness of the physical unknowns without requiring postprocessing or slope limiters;
iii) nonlinearities do not appear explicitly within differential operators or interface terms, giving the method with a natural parallelizable structure and high efficiency;
iv) it respects a discrete version of the entropy stability estimate of the continuous problem.

This is a joint work with Sergio Gómez and Ansgar Jüngel.

Speaker: Ilaria Perugia (Universität Wien)

15:15 → 16:00 Martin W. Licht - TBA

Speaker: Martin W. Licht (EPFL)

16:00 → 16:30 Coffee break

16:30 → 17:15 A posteriori goal-oriented error estimators based on equilibrated flux and potential reconstructions

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 use of some energy-norm error estimators.

In this talk, an overview of such techniques in different contexts will be given. We will then provide an upper-bound of the error which can be totally and explicitly computed for various discretization schemes. Finally, the behaviour of such estimators on some numerical benchmarks will be investigated. Two models will be particularly considered : a reaction-diffusion problem, and an eddy-current problem, arising in the context of low-frequency electromagnetisms.

Speaker: Emmanuel Creusé (Université Polytechnique Hauts-de-France)

17:15 → 18:00 Francesca Rapetti - TBA

Speaker: Francesca Rapetti (Université Côte d'Azur)

Tuesday, 11 June

09:30 → 10:15 Hybrid compatible Finite Element and Finite Volume discretization for viscous and resistive MHD

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, we developed a method based both on Finite Volumes for handling the slow convection and robust for shocks and on the other hand Finite Element Exterior Calculus which enable exact conservation of the main invariant of the system.

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

10:15 → 10:45 Coffee break

10:45 → 11:30 The Exterior Calculus Discrete De Rham complex

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 related to the topology of the domain. The importance of this complex and its properties in the stability analysis of models of partial differential equations (such as the Stokes/Navier-Stokes equations, magnetostatic equations, etc.) has been understood for decades. Reproducing the properties of this complex at the discrete level is essential for the design of stable schemes for these models, and is related, e.g., to the design of inf-sup stable methods for saddle point problems.

In the last two decades, the Finite Element Exterior Calculus (FEEC) framework has been set up to devise versions of the de Rham complex through the exterior calculus framework, which allows to treat all operators (gradient, curl, divergence) in a unified way as exterior derivatives of differential forms of certain degrees. These discrete complexes are however restricted to particular meshes (mostly made of tetrahedra and hexahedra), which do not easily lends themselves to standard scientific calculus techniques like local mesh refinement or mesh agglomeration (appearing, e.g., in multi-grid methods).

In this talk, we will present the Exterior Calculus Discrete De Rham (ECDDR) method. This is a discrete version of the de Rham complex of differential forms, that can be applied on polytopal meshes (made of generic polygons in 2D, generic polyhedra in 3D). As many polytopal methods, its design is based on adopting a higher and systematic view, which not only relaxes the conditions on the meshes, but can also lead to leaner methods than standard Finite Element methods. The design of ECDDR relies on the Stokes formula, which identifies the relevant degrees of freedom, as well as provides expressions for the discrete differential forms and potential reconstructions. We will show that the algebraic properties of the de Rham complex are preserved at the discrete level (including its cohomology), and we will explain how the tools in the ECDDR can be used to design numerical schemes.

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

11:30 → 12:15 Polytopal methods on Riemannian manifolds

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 and the differential operator,
such as the Nédélec or the Raviart-Thomas elements relating the curl operator to the circulation along edges, and the divergence operator to the flux across surfaces.
The connection between the usual differential operators (gradient, curl, and divergence) and the geometry
can be better seen through framework of the exterior calculus,
where these operators are unified as the exterior derivative applied to differential forms of different degrees.
The associated finite element spaces also have a natural description
which has been developed into the Finite Element Exterior Calculus (FEEC) framework,
leading to an intrinsic definition common to each space and operator in any dimension.

Several other methods that replicate the complexes structuring systems of differential equations at the discrete level were then developed
to use smoother spaces, different complexes, or more general meshes such as the Discrete De Rham (DDR) method.

Although the notions involving exterior calculus are manifestly independent of the underlying metric,
most discrete methods must ultimately assume a trivial space for the notion of simplicial/polytopal mesh, and for the notion of polynomial.
In this talk, we will present the generalisation of the Exterior Calculus Discrete De Rham (ECDDR) method to general Riamannian manifolds.
This construction uses a much more lenient notion of mesh, allowing to consider manifolds described by several charts,
and to use potentially any shape for the elements.
In particular, it is possible to use curved elements even when working on a flat space.
The basis functions are intrinsically defined element-wise. They are based on polynomial spaces of arbitrary order and are adapted to the chart and the metric.
We will then present a numerical application of this method to the Maxwell equations on a surface.

Speaker: Marien-Lorenzo Hanot (The University of Edinburgh)

12:15 → 14:30 Lunch break - W@else restaurant

14:30 → 15:15 Learning based reduction methods in the context of PDE constrained optimization

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 ap-
proach is the reduced basis method that relies on the approximation of the solu-
tion manifold of a parameterized system by tailored low dimensional approxima-
tion spaces that are spanned from suitably selected particular solutions, called
snapshots. With speedups that can reach several orders of magnitude, reduced
basis methods enable high fidelity real-time simulations for certain problem
classes and dramatically reduce the computational costs in many-query appli-
cations. While the ”online efficiency” of these model reduction methods is very
convincing for problems with a rapid decay of the Kolmogorov n-width, there
are still major drawbacks and limitations. Most importantly, the construction
of the reduced system in a so called ”offline phase” is extremely CPU-time and
memory consuming for large scale systems. For practical applications, it is thus
necessary to derive model reduction techniques that do not rely on a classical
offline/online splitting but allow for more flexibility in the usage of computa-
tional resources. In this talk we focus on learning based reduction methods in
the context of PDE constrained optimization and inverse problems and evaluate
their overall efficiency. We discuss learning strategies, such as adaptive enrich-
ment as well as a combination of reduced order models with machine learning
approaches in the contest of time dependent problems. Concepts of rigorous cer-
tification and convergence will be presented, as well as numerical experiments
that demonstrate the efficiency of the proposed approaches.

Speaker: Mario Ohlberger (Universität Münster)

15:15 → 16:00 Virginie Ehrlacher - TBA

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

16:00 → 16:45 Runge-Kutta methods are stable

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 systems of ODEs.
We explain the failure of different approaches, offer a new stability theory and demonstrate a few examples.

Speaker: Eitan Tadmor (University of Maryland)

16:45 → 17:15 Coffee break

17:15 → 19:30 Poster session

Margherita Castellano (Ecole Polytechnique)
Jean Cauvin-Vila (TU Vienna)
Farah Chaaban (ENSTA Paris)
Amélie Dupouy (Inria Lille)
Youssef Essadaoui (Université Sultan Moulay Slimane)
Maxime Jonval (Inria Lille, IFPEN)
Tino Laidin (Université de Lille)
Christina Mahmoud (Université de Montpellier)
Ismail Merabet (Kasdi Merbah University)
Julien Moatti (TU Vienna)
Jia Jia Qian (Monash University)
Marwa Salah (Université de Montpellier)

Wednesday, 12 June

09:30 → 10:15 Modeling Multiphase Multicomponent Porous Flows

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 and free energy (or
entropy) of the fluids will be highlighted. The construction of
numerical schemes which are robust in the presence of degeneracy, and
solution techniques which exploit the saddle point structure, will be
considered.

Speaker: Noel J. Walkington (Carnegie Mellon University)

10:15 → 10:45 Coffee break

10:45 → 11:30 Céline Grandmont - TBA

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

11:30 → 12:15 Geneviève Dusson - TBA

Speaker: Geneviève Dusson (CNRS, Université Bourgogne Franche-Comté)

12:15 → 14:30 Lunch break - W@else restaurant

14:30 → 15:15 Francis Filbet - TBA

Speaker: Francis Filbet (Université Paul Sabatier)

15:15 → 16:00 Emmanuel Franck - TBA

Speaker: Emmanuel Franck (Inria Centre at Université de Lorraine)

16:00 → 16:30 Coffee break

16:30 → 17:15 Khaled Saled - TBA

Speaker: Khaled Saleh (Université Claude Bernard, Lyon 1)

17:15 → 18:00 Andrea Thomann - TBA

Speaker: Andrea Thomann (Inria Centre at Université de Lorraine)

20:00 → 23:00 Social dinner - L'assiette du marché

Thursday, 13 June

09:30 → 10:15 Designing conservative and accurately dissipative numerical integrators in time

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 structure-preserving discretizations for
nonlinear problems, and in conserving non-polynomial invariants.

In this work we propose a new, general framework for the construction of
structure-preserving timesteppers via finite elements in time and the systematic
introduction of auxiliary variables. The framework reduces to Gauss methods
where those are structure-preserving, but extends to generate arbitrary-order
structure-preserving schemes for nonlinear problems, and allows for the
construction of schemes that conserve multiple higher-order invariants. We
demonstrate the ideas by devising novel schemes that exactly conserve all known
invariants of the Kepler and Kovalevskaya problems, high-order energy-conserving
and entropy-dissipating schemes for the compressible Navier–Stokes equations,
and multi-conservative schemes for the Benjamin-Bona-Mahony equation.

Speaker: Patrick E. Farrell (University of Oxford)

10:15 → 10:45 Coffee break

10:45 → 11:30 Dietmar Gallistl - TBA

Speaker: Dietmar Gallistl (Universität Jena)

11:30 → 12:15 Iain Smears - TBA

Speaker: Iain Smears (University College London)

12:15 → 14:00 Workshop closing / Goodbye buffet