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Efficient Numerical Schemes for Evolution Equations with Singularities and Shocks.

Zuodong Wang: Thursday, 18th Sep 2025 – 10h30 to 12h00 Abstract: In this Thesis, we develop efficient numerical schemes for evolution partial differential equations (PDEs) with shocks and singularities. Such PDEs arise in diverse applications, including fluid dynamics, phase transition, and radiation transport. In Chapter 3, we solve the Yee–LeVeque model and its generalization using a novel numerical scheme that perturbs the stiff reaction parameter in a mesh-dependent manner. For this scheme, we establish a maximum principle and entropy inequalities without restrictive assumptions on the mesh-size or the time-step. In Chapter 4, we study the steady neutron transport equation with stiff reaction terms. By applying the edge stabilization technique, we relax some mesh constraints and develop an efficient post-processing technique to remove non-physical oscillations while preserving the positivity and conservation properties of the scheme. In Chapter 5, we focus on the Allen–Cahn equation. A fundamental phase-field model with stiff source terms. Here, we propose a nonlinear scheme for which we prove a maximum principle and derive an optimal energy-error bound with polynomial dependence on the singularity perturbation factor (or reaction parameter). In Chapter 6, we investigate the linear wave equation. Our contributions are novel $hp$-a priori and a posteriori error bounds, valid even for low-regularity solutions, and an adaptive time refinement scheme driven by our a posteriori error estimators. Numerical experiments across all chapters demonstrate the efficiency and reliability of the proposed schemes.

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Stabilisation of the high-order discretised wave equation for data assimilation problems

Sébastien Imperiale: Thursday, 16th Oct 2025 – 10h30 to 12h00 Abstract: The objective of this work is to propose and analyze numerical schemes to solve transient wave propagation problems that are exponentially stable (i.e. the solution decays to zero exponentially fast). Applications are in data assimilation strategies or the discretisation of absorbing boundary conditions. More precisely the aim of our work is to propose a discretization process that enables to preserve the exponential stability at the discrete level as well as a high order consistency when using a high-order finite element approximation. The main idea is to add to the wave equation a stabilizing term which damps the high-frequency oscillating components of the solutions such as spurious waves. This term is built from a discrete multiplier analysis that proves the exponential stability of the semi-discrete problem at any order without affecting the order of convergence.

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Model order reduction for parametrized PDEs: An introduction & recent advances in the application to Friedrichs’ systems

Lukas Renelt: Thursday, 24th July 2025 – 10h30 to 12h00 Abstract: The numerical solution of partial differential equations (PDEs) is one of the main research fields in computational mathematics where a vast variety of numerical methods have been developed. Applications include the simulation of reactive transport, groundwater flow, electromagnetism or even the computation of quantum states. These equations often additionally depend on physical parameters such as coefficients or boundary data which can significantly influence the solutions behaviour. This poses an additional challenge if solutions for many different parameter values are required such as in parameter studies, optimization tasks, inverse problems or uncertainty quantification. While methods such as the finite element method, finite volumes or discontinuous Galerkin approaches work well for given fixed parameters, they are prohibitively costly if solutions for thousands of different parameters are needed. To address this challenge, model order reduction methods have been developed which aim at approximating the highly complex set of all possible solutions jointly by a small (linear) subspace. In this talk, we will give a general introduction to the methodology with particular focus on the Reduced Basis (RB) approach highlighting both the abstract analysis and also showing concrete realizations. In the second part of the talk, we will present recent results when applying the method to parametrized Friedrichs’ systems – a large abstract class of linear PDE problems including for example convection-diffusion-reaction, linear transport, linear elasticity or the time-harmonic Maxwell equations. From a theoretical point of view, these problems are particularly interesting as their Friedrichs’ formulation involves parameter-dependent function spaces – a setting which has not been explored by the model order reduction community thus far. We present a novel theoretical framework and highlight the connections to the established theory with implications beyond Friedrichs’ systems. Additionally, a normal-equation-based discretization is introduced and used…

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A Robust Linearization Method for Complementarity Problems: A Detour Through Hyperplane Arrangements

Baptiste Plaquevent-Jourdain: Monday, 7th July 2025 – 10h30 to 12h00 Abstract: The initial goal of this thesis is the resolution of complementarity problems. These problems are reformulated here by the minimum C-function, which is piecewise thus nondifferentiable and leads to nonsmooth systems. The globalization of local methods for such equations (semi smooth Newton for instance) generally faces the difficulty that the computed directions are not necessarily descent direction for the associated merit function, used in lineasearch methods (whereas for smooth equations, the opposite of the gradient always works). In the case of the minimum C-function, a recent method replaces the pseudo-linearization direction by a direction found in a suitable convex polyhedron, guaranteed to be nonempty by some stringent regularity condition. The initial objective was to remove the regularity condition, as for the smooth systems, by using a Levenberg-Marquardt approach. The piecewise smooth aspect of the merit function induced by the minimum implies to choose a certain piece, this choice being discussed towards the end of the presentation. When trying to better understand this method and the generalization of the derivative, the B(ouligand)-differential, for the minimum function, it appeared that, for the simple case of linear problems (or affine), the inherent structure of the B-differential is the one of an arrangement or hyperplanes. This problem, very classic in combinatorial geometry, that we discovered here, is actually surprisingly rich and deep. We propose improvements on a state-of-the-art algorithm identifying the chambers. In particular, « primal-dual » variants, linking the chambers of an arrangement with the circuits of the associated matroid, seem promising. This long detour is actually relevant for the nonsmooth method and the choice of the « piece ».

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Hybrid high-order methods for the numerical simulation of elasto-acoustic wave propagation

Romain Mottier: Thursday, 17th July 2025 – 10h30 to 12h00 Abstract: The objective of this Thesis is to develop approximation methods for the simulation of elasto-acoustic wave propagation. The spatial discretization of these equations relies on Hybrid High-Order (HHO) methods, which offer several advantageous properties, such as the ability to handle general meshes, computational efficiency, and high accuracy. An energy-norm error estimate is derived in the space semi-discrete setting. The time discretization is based on Runge–Kutta schemes. Particular attention is given to the stability of explicit schemes and to the efficiency of both explicit and implicit approaches. Numerical results are presented, including a geophysical application involving a complex geometry and strong heterogeneities. A second important aspect explored in this Thesis concerns the use of so-called unfitted meshes for elliptic problems with curved interfaces. These meshes do not conform to the physical interfaces, leading to mesh cells that may be intersected by them. This approach simplifies the mesh design. A method based on polynomial extension is introduced and analyzed in order to handle ill-cut cells. This approach is less intrusive than the more classical one based on the agglomeration of ill-cut cells.

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