Section 4 Seminar, Spring 2026

This talk addresses sharpness conditions for constructing $C^r$ conforming finite element spaces from superspline spaces on general simplicial triangulations. We introduce the concept of extendability for the pre-element spaces, which encompasses both the superspline spaces and the finite element spaces. By examining the extendability condition for both types of spaces, we provide an answer to the conditions regarding the construction. A corollary of our results is that constructing $C^{r}$ conforming elements in $d$ dimensions generally requires an extra $C^{2^{s - 1} r}$ continuity on $s$-codimensional subsimplices, and the polynomial degree is at least $(2^d r + 1)$.

The talk will focus on an evolutionary PDE modelling a one-dimensional linearly elastic body interacting with a rigid substrate through an adhesive material. More precisely, the PDE is a damped semilinear wave equation with Neumann boundary conditions. The main feature of the equation is the nonlinear force acting on the body: it is proportional to the displacement for small displacements and it is zero when the displacement is large.

The talk will be divided in three parts. In the first part, we will recall some key aspects of the classical theory of maximal monotone operators to deduce the global well-posedness for the model. In the second part, we will focus on the qualitative asymptotic behaviour of solutions for large time, showing that solutions converge to solutions of the steady problem. Finally, in the third part, we will show how to deduce a quantitative estimate on the rate of convergence to the asymptotic limit.

The content of the talk is based on a paper in collaboration with N. De Nitti, F. Maddalena, G. Orlando, and E. Zuazua.

Magnetohydrodynamics (MHD) models electrically conducting fluids such as plasmas. The numerical approximation of MHD poses significant challenges due to its nonlinear hyperbolic structure, the presence of involution constraints, and the need to preserve key physical invariants such as positivity and entropy inequalities. In this talk, we present recent results on the design and analysis of structure- and invariant domain-preserving finite element methods for the compressible MHD equations. At the PDE level, we introduce viscous parabolic regularizations that are compatible with all generalized entropies, satisfy a minimum entropy principle, and preserve the positivity of density and internal energy. These results provide a robust foundation for the numerical approximation of MHD. Motivated by these analytical developments, we then propose a structure-preserving finite element method based on operator splitting. The method combines a compressible Euler solver with a curl-conforming discretization of the magnetic field, ensuring exact preservation of involution constraints. We show that the scheme satisfies invariant domain properties, including positivity and entropy stability, and that it inherits conservation and stability properties from the underlying Euler solver. A series of challenging MHD benchmark problems is used to assess the performance of the method, demonstrating both high-order accuracy for smooth solutions and robust shock-capturing capabilities for nonsmooth regimes. The talk is mainly based on the following recently published papers: 1. Tuan Anh Dao, Lukas Lundgren, Murtazo Nazarov. Viscous regularization of the MHD equations. (2024); SIAM Journal on Applied Mathematics. 84 (4): 1439–1459 2. Tuan Anh Dao, Murtazo Nazarov, Ignacio Tomas. A structure-preserving numerical method for the ideal compressible MHD system. (2024); Journal of Computational Physics. 508: 113009 3. Tuan Anh Dao, Murtazo Nazarov. Monolithic parabolic regularization of the MHD equations and entropy principles. (2022); Computer Methods in Applied Mechanics and Engineering. 398: 115269

The inverse scattering transform (IST) method, reformulated as a Riemann–Hilbert (RH) factorization problem, provides a powerful analytical framework for solving nonlinear integrable evolution equations. Extending this approach from the real line to finite-interval or periodic settings introduces additional challenges due to the presence of boundary data. These difficulties can be addressed by the Unified Transform Method (also known as the Fokas Method), which simultaneously analyzes both equations of the Lax pair and connects the associated spectral functions through a global relation. For certain boundary conditions, referred to as linearizable, the global relation can be resolved explicitly, allowing the formulation of a Riemann–Hilbert problem in terms of given data alone. In this talk, we present the application of this framework to the Camassa–Holm equation on a finite interval with periodic boundary conditions. We show that this problem is linearizable and that its solution can be obtained in the form of a RH factorization problem whose data are determined by the scattering matrix associated with the initial data.